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THE SCIENCE OF MECHANICS

THE

SCIENCE OF MECHANICS

A CRITICAL AND HISTORICAL EXPOSITION

OF ITS PRINCIPLES

■ BY

DR. ERNST MACH
Propbssor op Physics in thb Universitv op Pracub

TRANSLATED FROM THE SECOND GERMAN EDITION

THOMAS J. McCORMACK

% BY

Wrm TWO HUNDRBD AND PIPTY CUTS AND ILLUSTRATIONS

CHICAGO
THE OPEN COURT PUBLISHING CO.

1893

Thb Opkm Coukt Pubushinc Ca

• • •

V\'

fit' Littcsific ^irsf

R. R. DONNELLEY * SONS CO.. CHICAGO

TRANSLATOR'S PREFACE.

ERRATA.

• Page 140. — Lines i and 2 from top, for its perptndicular read the vertical.

Page 14a. — Last line, read no force for a force.

Page 277.— Between the third and fourth lines from the bottom insert the
following: "employed as the anit of mass, an arbitrary length is."

Page 283 — Line 13 from bottom, read jr^^o/A for S26oth.

Page 337. — Line 9 from bottom, for perpendicular read vertical.

Page 398.— Line 5 from bottom, read Xdx for Xdz.

all the proofs and has rewritten § 8 in the chapter on
Units and Measures, where the original was inappli-
cable to this country and slightly out of date.

Thomas J. McCormack.
La Salle, III., June 28, 1893.

R. R. DONNELLEY * SONS CO.. CHICAGO

TRANSLATOR'S PREFACE.

The Open Court Publishing Company has*acquired
the sole right of English translation of this work,
which in its German original formed a volume of
the Internationale wissenschaftliche Bibliothek, of F. A.
BrockhauSy of Leipsic.

In the reproduction, many textual errors and ir-
regularities have been corrected, marginal titles have
been inserted, and the index has been amplified. It
is believed that the usefulness of the book has thus
been increased.

No pains have been spared to render the author's
meaning clearly and faithfully. In this, it has often
been necessary to depart widely from the form of the
original, but never, it is hoped, from its spirit.

The thanks of the translator are due to Mr. C. S.
Peirce, well known for his studies both of analytical
mechanics and of the history and logic of physics, for
numerous suggestions and notes. Mr. Peirce has read
all the proofs and has rewritten § 8 in the chapter on
Units and Measures, where the original was inappli-
cable to this country and slightly out of date.

Thomas J. McCormack.

La Salle, III., June 28, 1893.

AUTHOR'S PREFACE TO THE TRANS-
LATION.

■

Having read the proofs of the present translation
of my work, Die Mechanik in ihrer Eniwickelungy I can
testify that the publishers have supplied an excellent,
accurate, and faithful rendering of it, as their previous
translations of essays of mine gave me every reason to
expect. My thanks are due to all concerned, and
especially to Mr. McCormack, whose intelligent care
in the conduct of the translation has led to the dis-
covery of many errors, heretofore overlooked. I may,
thus, confidently hope, that the rise and growth of the
ideas of the great inquirers, which it was my task to
portray, will appear to my new public in distinct and
sharp outlines. E. Mach.

Prague, April 8th, 1893.

PREFACE TO THE FIRST EDITION.

■

The present volume is not a treatise upon the ap-
plication of the principles of mechanics. Its aim is
to clear up ideas, expose the real significance of the
matter, and get rid of metaphysical obscurities. The
little mathematics it contains is merely secondary to
this purpose.

Mechanics will here be treated, not as a branch of
mathematics, but as one of the physical sciences. If
the reader's interest is in that side of the subject, if he
is curious to know how the principles of mechanics
have been ascertained, from what sources they take
their origin, and how far they can be regarded as per-
manent acquisitions, he will find, I hope, in these
pages some enlightenment. All this, the positive and
physical essence of mechanics, which makes its chief
and highest interest for a student of nature, is in exist-
ing treatises completely buried and concealed beneath
a mass of technical considerations.

The gist and kernel of mechanical ideas has in al-
most every case grown up in the investigation of very
simple and special cases of mechanical processes ; and
the analysis of the history of the discussions concern-

VIII PREFACE TO THE FIRST EDITION.

ing these cases must ever remain the method at once
the most effective and the most natural for laying this
gist and kernel bare. Indeed, it is not too much to
say that it is the only way in which a real comprehen-
sion of the general upshot of mechanics is to be at-
tained.

I have framed my exposition of the subject agree-
ably to these views. It is perhaps a little long, but, on
the other hand, I trust that it is clear. I have not in
every case been able to avoid the use of the abbrevi-
ated and precise terminology of mathematics. To do
so would have been to sacrifice matter to form \ for the
language of everyday life has not yet grown to be suf-
ficiently accurate for the purposes of so exact a science
as mechanics.

The elucidations which I here offer are, in part,
substantially contained in my treatise, Die Geschichte
und die Wurzel des Satzes von der Erhaltung der Arbeit
(Prague, Calve, 1872). At a later date nearly the same
views were expressed by Kirchhoff ( Vorlesungen Uber
mathematische Physik: Mechanik^ Leipsic, 1874) and by
Helmholtz ^Die Thatsachen in der Wahrnehmung^
Berlin, 1879), and have since become commonplace
enough. Still the matter, as I conceive it, does not
seem to' have been exhausted, and I cannot deem my
exposition to be at all superfluous.

In my fundamental conception of the nature of sci-
ence as Economy of Thought, — a view which I in-
dicated both in the treatise above cited and in my

PREFACE TO THE FIRST EDITION, ix

pamphlet, Die Gestalten der Flussigkeit (Prague, Calve,
1872), and which I somewhat more extensively devel-
oped in my academical memorial address. Die okono-
mische Natur der physikalischen Forschung (Vienna, Ce-
roid, 1882, — I no longer stand alone. I have been
much gratified to find closely allied ideas developed,
in an original manner, by Dr. R. Avenarius {Philoso-
phie als Denken der Welt, gerndss detn Princip des klein-
sten Kra/tmaasses, Leipsic, Fues, 1876). Regard for
the true endeavor of philosophy, that of guiding into
one common stream the many rills of knowledge, will
not be found wanting in my work, although it takes a
determined stand against the encroachments of meta-
physical methods.

The questions here dealt with have occupied me
since my earliest youth, when my interest for them was
powerfully stimulated by the beautiful introductions of
Lagrange to the chapters of his Analytic Mechanics, as
well as by the lucid and lively tract of Jolly, Principien
der Mechanik (Stuttgart, 1852). If Duehring's esti-
mable work, Kritische Geschichte der Principien der Me-
chanik (Berlin, 1873), did not particularly influence
me, it was that at the time of its appearance, my ideas
had been not only substantially worked out, but actually
published. Nevertheless, the reader will, at least on
the destructive side, find many points of agreement
between Dtihring's criticisms and those here expressed.

The new apparatus for the illustration of the sub-
ject, here figured and described, were designed entirely

X PREFACE TO THE FIRST EDITION.

by me and constructed by Mr. F. Hajek, the mechani-
cian of the physical institute under my control.

In less immediate connection with the text stand
the fac-simile reproductions of old originals in my pos-
session. The quaint and naive traits of the great in-
quirers, which find in them their expression, have al-
ways exerted upon me a refreshing influence in my
studies, and I have desired that my readers should
share this pleasure with me.

E. Mach.

Prague, May, 1883.

PREFACE TO THE SECOND EDITION.

In consequence of the kind reception which this
book has met with, a very large edition has been ex-
hausted in less than iive years. This circumstance and
the treatises that have since then appeared of E. Wohl-
will, H. Streintz, L. Lange, J. Epstein, F. A. Miiller,
J. Popper, G. Helm, M. Planck, F. Poske, and others
are evidence of the gratifying fact that at the present
day questions relating to the theory of cognition are
pursued with interest, which twenty years ago scarcely
anybody noticed.

As a thoroughgoing revision of my work did not
yet seem to me to be expedient, I have restricted my-
self, so far as the text is concerned, to the correction
of typographical errors, and have referred to the works
that have appeared since its original publication, as
far as possible, in a few appendices.

E. Mach.
Prague, June, 1888.

TABLE OF CONTENTS.

PAOB

Translator's Preface v

Anthor's Preface to the Translation vi

Preface to the First Edition vii

Preface to the Second Edition xi

Table of Contents xiii

Introduction I

CHAPTER I.

THE DEVELOPMENT OF THE PRINCIPLES OF STATICS.

I. The Principle of the Lever 8

II. The Principle of the Inclined Plane 24

III. The Principle of the Composition of Forces . . . . 33

IV. The Principle of Virtual Velocities 49

V. Retrospect of the Development of Statics 77

VI. The Principles of Statics in Their Application to Fluids 86

VII. The Principles of Statics in Their Application to Gas-

eous Bodies no

CHAPTER n.

THE DEVELOPMENT OF THE PRINCIPLES OF DYNAMICS.

I. Galileo's Achievements 128

n. The Achievements of Huygens . . 155

III. The Achievements of Newton 187

IV. Discussion and Illustration of the Principle of Reaction 201
V. Criticism of the Principle of Reaction and of the Con-
cept of Mass 216

VI. Newton's Views of Time, Space, and Motion . . . 222

VII. Synoptical Critique of the Newtonian Enunciations. . 238

VIII. Retrospect of the Development of Dynamics .... 245

XIV TABLE OF CONTENTS,

CHAPTER III.

THE EXTENDED APPLICATION OF THE PRINCIPLES OF
MECHANICS AND THE DEDUCTIVE DEVELOP-
MENT OF THE SCIENCE.

PAGiE

I. Scope of the Newtonian Principles 256

II. The Formulae and Units of Mechanics 269

III. The Laws of the Conservation of Momentum, of the

Conservation of the Centre of Gravity, and of the

Conservation of Areas 287

IV. The Laws of Impact 305

V. D'AIembert's Principle 331

VI The Principle of Vis Viva 343

VII. The Principle of Least Constraint 350

VIII. The Principle of Least Action 364

IX. Hamilton's Principle ... 380

X. Some Applications of the Principles of Mechanics to

Hydrostatic and Hydrodynamic Questions . . . .384

CHAPTER IV.

THE FORMAL DEVELOPMENT OF MECHANICS.

I. The Isoperimetrical Problems 421

II. Theological, Animistic, and Mystical Points of View

in Mechanics 446

III Analytical Mechanics . 465

IV. The Economy of Science 481

CHAPTER V.

THE RELATION OF MECHANICS TO OTHER DEPART-
MENTS OF KNOWLEDGE.

I The Relations of Mechanics to Physics 495

II. The Relations of Mechanics to Physiology . . . .504

Appendix 509

Chronological Table of a Few Eminent Inquirers and of Their

More Important Mechanical Works 519

Index . . .• 523

THE SCIENCE OF MECHANICS

INTRODUCTION.

1. That branch of physics which is at once the old- The science
est and the simplest and which is therefore treated ics.

as introductory to other departments of this science,
is concerned with the motions and equilibrium of
masses. It bears the name of mechanics.

2. The history of the development of mechanics,
is quite indispensable to a full comprehension of the
science in its present condition. It also affords a sim-
ple and instructive example of the processes by which
natural science generally is developed.

An instinctive J irreflective knowledge of the processes instinctive
of nature will doubtless always precede the scientific,
conscious apprehension, or investigation, of phenom-
ena. The former is the outcome of the relation in
which the processes of nature stand to the satisfac-
tion of our wants. The acquisition of the most ele-
mentary truth does not devolve upon the individual
alone : it is pre-effected in the development of the race.

In point of fact, it is necessary to make a dis- Mechanical
tinction between mechanical experience and mechan- ®*p®"*°^®*
ical science, in the sense in which the latter term is at
present employed. Mechanical experiences are, un-
questionably, very o\d. If we carefully examine the
ancient Egyptian and Assyrian monuments, we shall
find there pictorial representations of many kinds of

a THE SCIENCE OF MECHANICS.

Tbcme- implements and mechanical contrivances; but ac-
knoviad^s counts of the scientific knowledge of these peoples
are either totally lacking, or point conclusively to a
very inferior grade of attainment. By the side of
highly ingenious ap-
pliances, we behold
the crudest and rough-
est expedients em-
ployed— as the use of
sleds, for instance, for
the transportation of
enormous blocks of
stone. All bears an
instinctive, un perfec-
ted, accidental char-
acter.

So, too, prehistoric
graves contain imple-
^ mentswhoseconstruc-

f" tion and employment

imply no little skill
and much mechanical
experience. Thus, long
before theory was
dreamed of, imple-
ments, machines, me-
chanical experien-
ces, and mechanical
knowledge were abun-

Hbvcih 3. The idea often

ill suggests Itself that

perhaps the incom-
plete accounts we pos-

INTRODUCTION. 3

sess have led us to underrate the. science of the ancient
world. Passages occur in ancient authors which seem
to indicate a profounder knowledge than we are wont
to ascribe to those nations. Take, for instance, the
following passage from Vitruvius, ^ i>^ Architectural
Lib. V, Cap. Ill, 6 ; L. -^ .^. ^ <^ >ci ^

The voice is a flowing breath, made sensible to a passage

from Vitm-
vius.

** the organ of hearing by the movements it produces
'<in the air. It is propagated in infinite numbers of
** circular zones: exactly as when a stone is thrown
*'into a pool of standing water countless circular un-
'^dulations are generated therein, which, increasing

* * as they recede from the centre, spread out over a
** great distance, unless the narrowness of the locality
**or some obstacle prevent their reaching their ter-

* * mination ; for the first line of waves, when impeded
*'by obstructions, throw by their backward swell the
•* succeeding circular lines of waves into confusion.
" Conformably to the very same law, the voice also
*' generates circular motions ; but with this distinction,
** that in water the circles, remaining upon the surface,
** are propagated in the horizontal direction only, while

* * the voice is propagated both horizontally and ver-
**tically."

Does not this sound like the imperfect exposition controvert-
of a popular author, drawn from more accurate disqui- evidencc.**^
sitions now lost ? In what a strange light should we
ourselves appear, centuries hence, if our popular lit-
erature, which by reason of its quantity is less easily
destructible, should alone outlive the productions of
science ? This too favorable view, however, is very
rudely shaken by the multitude of other passages con-
taining such crude and patent errors as cannot be con-
ceived to exist in any high stage of scientific culture.

4 THE SCIENCE OF MECHANICS,

The origin 4. When, where, and in what manner the develop-
ment of science actually began, is at this day difficult
historically to determine. It appears reasonable to
assume, however, that the instinctive gathering of ex-
periential facts preceded the scientific classification of
them. Traces of this process may still be detected in
the science of to-day; indeed, they are to be met with,
now and then, in ourselves. The experiments that
man heedlessly and instinctively makes in his strug-
gles to satisfy his wants, are just as thoughtlessly and
unconsciously applied. Here, for instance, belong the
primitive experiments concerning the application of
the lever in all its manifold forms. But the things
that are thus unthinkingly and instinctively discovered,
can never appear as peculiar, can never strike us as
surprising, and as a rule therefore will never supply an
impetus to further thought.
Thefunc- The transition from this stage to the classified,

da? ciassM Scientific knowledge and apprehension of facts, first be-
veioi>ment comes possible on the rise of special classes and pro-
fessions who make the satisfaction of definite social
wants their lifelong vocation. A class of this sort oc-
cupies itself with particular kinds of natural processes.
The individuals of the class change ; old members
drop out, and new ones come in. Thus arises a need
of imparting to those who are newly come in, the
stock of experience and knowledge already possessed ; 'la

a need of acquainting them with the conditions of the ) I n

ill

The com- attainment of a definite end so that the result may be ( I w

of knowi- determined beforehand. The communication of knowl- ' I ni

edge is thus the first occasion that compels distinct re- es

flection, as everybody can still observe in himself. tk

Further, that which the old members of a guild me- . i^

chanically pursue, strikes a new member as unusual I W(

(

INTRODUCTION, ^ 5

and strange, and thus an impulse is given to fresh re-
flection and investigation.

When we wish to bring to the knowledge of a per- involves

dotcriptioiL

son any phenomena or processes of nature, we have
the choice of two methods : we may allow the person to
observe matters for himself, when instruction comes
to an end ; or, we may describe to him the phenomena
in some way, so as to save him the trouble of per-
sonally making anew each experiment. Description,
however, is only possible of events that constantly re-
cur, or of events that are made up of component
parts that constantly recur. That only can be de-
scribed, and conceptually represented which is uniform
and conformable to law ; for description presupposes
the employment of names by which to designate its
elements ; and names can acquire meanings only when
applied to elements that constantly reappear.

5. In the infinite variety of nature many ordinary a unitary

•^ . ■' -^ ■' conception

events occur; while others appear uncommon, per- of nature,
plexing, astonishing, or eVen contradictory to the or-
dinary run of things. As long as this is the case we
do not possess a well-settle4 and unitary conception of
nature. Thence is imposed the task of everywhere
seeking out in the natural phenomena those elements
that are the same, and that amid all multiplicity are
ever present. By this means, on the one hand, the
most economical and briefest description and com-
munication are rendered possible; and on the other. The nature
when once a person has acquired the skill of recog-edge.
nising these permanent elements throughout the great-
est range and variety of phenomena, of seeing them in
the same, this ability leads to a comprehensive^ compact^
consistent f and facile conception of the facts. When once
we have reached the point where we are everywhere

6 THE SCIENCE OF MECHANICS.

Theadap- able to detect the same few simple elements, combin-

tation of . , ,

thoughts to ing in the ordinary manner, then they appear to us as
things that are familiar ; we are no longer surprised,
there is nothing new or strange to us in the phenom-
ena, we feel at home with them, they no longer per-
plex us, they are explained. It is a process of adaptation
of thoughts to facts with which we are here concerned.
The ccon- 6. Economy of communication and of apprehen-
tbooght. sion is of the very essence of science. Herein lies
its pacificatory, its enlightening, its refining element.
Herein, too, we possess an unerring guide to the his-
torical origin of science. In the beginning, all economy
had in immediate view the satisfaction simply of bodily
wants. With the artisan, and still more so with the
investigator, the concisest and simplest possible knowl-
edge of a given province of natural phenomena — a
knowledge that is attained with the least intellectual
expenditure — naturally becomes in itself an econom-
ical aim ; but though it was at first a means to an end, !
when the mental motives connected therewith are once
developed and demand their satisfaction, all thought
of its original purpose, ths personal need, disappears.
Further de- To find, then, what remains unaltered in the phe-
pf these nomena of nature, to discover the elements thereof

ideas*

and the mode of their interconnection and interdepend-
ence— this is the business of physical science. It en-
deavors, by comprehensive and thorough description,
to make the waiting for new experiences unnecessary ; '
it seeks to save us the trouble of experimentation, by
making use, for example, of the known interdepend-
ence of phenomena, according to which, if one kind of
event occurs, we may be sure beforehand that a certain
other event will occur. Even in the description itself
labor may be saved, by discovering methods of de-

WTRODUCTIOA'. 7

scribing the greatest possible number of different ob- Their prea-

ent discus-

jects at once and in the concisest manner. All this will sion merely

.... preparatory

be made clearer by the examination of points of detail
than can be done by a general discussion. It is fitting,
however, to prepare the way, at this stage, for the
most important points of outlook which in the course
of our work we shall have occasion to occupy.

7. We now propose to enter more minutely into the proposed
subject of our inquiries, and, at the same time, without rreatment.
making the history of mechanics the chief topic of
discussion, to consider its historical development so
far as this is requisite to an understanding of the pres-
ent state of mechanical science, and so far as it does
not conflict with the unity of treatment of our main
subject. Apart from the consideration that we cannot
afford to neglect the great incentives that it is in our
power to derive from the foremost intellects of allxheincer-
epochs, incentives which taken as a whole are more rived from
fruitful than the greatest men of the present day are with the
able to offer, there is no grander, no more intellectually lects of the
elevating spectacle than that of the utterances of the
fundamental investigators in their gigantic power.
Possessed as yet of no methods, for these were first
created by their labors, and are only rendered compre-
hensible to us by their performances, they grapple with
and subjugate the object of their inquiry, and imprint
upon it the forms of conceptual thought. They that
know the entire course of the development of science,
will, as a matter of course, judge more freely and And the in-
more correctly of the significance of any present scien- power
tific movement than they, who limited in their views a contact
to the age in which their own lives have been spent,
contemplate merely the momentary trend that the course
of intellectual events takes at the present moment.

{

CHAPTER I.

' THE DEVELOPMENT OF THE PRINCIPLES OF

STATICS.

I.
THE PRINCIPLE OF THE LEVER.

The earliest I. The earliest investigations concerning mechan-

mecbanical . * i • i_ i_ i • • •

researches ics of which we have any account, the investigations
statics. of the ancient Greeks, related to statics, or to the doc-
trine of equilibrium. Likewise, when after the taking
of Constantinople by the Turks in 1453 a fresh impulse
was imparted to the thought of the Occident by the an-
cient writings that the fugitive Greeks brought with
them, it was investigations in statics, principally evoked
by the works of Archimedes, that occupied the fore-
most investigators of the period.
Archimedes 2. Archimedes of Syracuse (287-2 1 2 B. C.) left
(287-212 B. behind him a number of writings, of which several
have come down to us in complete form. We will
first employ ourselves a moment with his treatise De
^quiponderantibusy which contains propositions re-
specting the lever and the centre of gravity.

In this treatise Archimedes starts from the follow-
ing assumptions, which he regards as self-evident :
Axiomatic a. Magnitudes of equal weight acting at equal

assump- . ,

tions of Ar- distances (from their point of support) are in equi-

chimedes. i., •

hbnum.

THE PRINCIPLES OP STATICS. 9

b. Magnitudes of equal weight acting at une- Axiomatic
qual distances (irom their point of support) are dons of Ar-

. 'ft • « 1 1 chimedeft.

not m equihbrium, but the one acting at the
greater distance sinks.
From these assumptions he deduces the following
proposition :

c. Commensurable magnitudes are in equilib-
rium when they are inversely proportional to their
distances (from the point of support).

It would seem as if analysis could hardly go be-
hind these assumptions. This is, however, when we
carefully look into the matter, not the case.

Imagine (Fig. 2) a bar, the weight of which is
neglected. The bar rests on a fulcrum. At equal dis-
tances from the fulcrum we ap-
pend two equal weights. That 1 7^

the two weights, thus circum- rS
stanced, are in equilibrium, is
the assumption from which Archi-
medes starts. We might suppose that this was self- Analysis of
evident entirely apart from any experience, agreeably to medean as-
the so-called principle of sufRcient reason ; that in view '""P"°"'
of the symmetry of the entire arrangement there is no
reason why rotation should occur in the one direction
rather than in the other. But we forget, in this, that
a great multitude of negative and positive experiences
is implicitly contained in our assumption ; the negative,
for instance, that dissimilar colors of the lever-arms,
the position of the spectator, an occurrence in the vi-
cinity, and the like, exercise no influence ; the positive,
on the other hand, (as it appears in the second as-
sumption,) that not only the weights but also their dis-
tances from the supporting point are decisive factors
in the disturbance of equilibrium, that they also are cir-

lo THE SCIENCE OF MECHANICS.

cumstances determinative of motion. By the aid of
these experiences we do indeed perceive that rest (no
motion) is the only motion which can be uniquely* de-
termined, or defined, by the determinative conditions
of the case.f
Character Now we are entitled to regard our knowledge of

the Archi- the decisive conditions of any phenomenon as sufficient
suits. only in the event that such conditions determine the

phenomenon precisely and uniquely. Assuming the
fact of experience referred to, that the weights and
their distances alone are decisive, the first proposition
of Archimedes really possesses a high degree of evi-
dence and is eminently qualified to be made the foun-
dation of further investigations. If the spectator place
himself in the plane of symmetry of the arrangement
in question, the first proposition manifests itself, more-
over, as a highly imperative instinctive perception, — a
result determined by the symmetry of our own body.
The pursuit of propositions of this character is, fur-
thermore, an excellent means of accustoming ourselves
in thought to the precision that nature reveals in her
processes.
The general 3- We will now reproduce in general outlines the
of tfe leUr train of thought by which Archimedes endeavors to re-
the simple duce the general proposition of the lever to the par-
uUrcMe.^ ticular and apparently self-evident case. The two
equal weights i suspended at a and b (Fig. 3) are, if
the bar ab be free to rotate about its middle point r, in
equilibrium. If the whole be suspended by a cord at
r, the cord, leaving out of account the weight of the

* So as to leave only a single possibility open.

t If, for example, we were to assume that the weight at the right de-
scended, then rotation in the opposite direction also would be determined by
the spectator, whose person exerts no influence on the phenomenon, taking
np bis position on the opposite side.

THE PRINCIPLES OF STATICS.

bar, will have to support the weight 2. The e'qual The general
weights at the extremities of the bar supply accor- of £e lever
dingly the place of the double weight at the centre. the timpie

and partic-
ular case.

i b3

Fir. 3.

Fig. 4.

On a lever (Fig. 4), the arms of which are in the
proportion of i to 2, weights are suspended in the pro-
portion of 2 to I. The weight 2 we imagine replaced
by two weights i, attached on either side at a distance
I from the point of suspension. Now again we have
complete symmetry about the point of suspension, and
consequently equilibrium.

On the lever-arms 3 and 4 (Fig. 5) are suspended
the weights 4 and 3. The lever-arm 3 is prolonged
the distance 4, the arm 4 is prolonged the distance 3,
and the weights 4 and 3 are replaced respectively by

/:,.

-A^

"N/'

ti t t'^h h h h\h"h 5

Fig. 5

4 and 3 pairs of symmetrically attached weights J,
in the manner indicated in the figure. Now again we
have perfect symmetry. The preceding reasoning, The genep
which we have here developed with specific figures, is * *""°"*
easily generalised.

4. It will be of interest to look at the manner in
which Archimedes's mode of view, after the precedent
of Stevinus, was modified by Galileo.

THE SCIENCE OF MECHANICS,

2 m

2«

Fig. 6.

Galileo's 'Galileo imagines (Fig. 6) a heavy horizontal prism,

treatment, homogeneous in material composition, suspended by

its extremities from a homogeneous bar of the same

length. The bar is provided at its middle point

with a suspensory attach -
ment. In this case equi-
librium will obtain ; this
we perceive at once. But
in this case is contained
every other case. Which
Galileo shows in the
following manner. Let
us suppose the whole
length of the bar or the prism to be 2(»i -j- «)• Cut
the prism in two, in such a manner that one portion
shall have the length 2m and the other the length 2«.
We can effect this without disturbing the equilibrium
by previously fastening to the bar by threads, close to
the point of proposed section, the inside extremities of
the two portions. We may then remove all the threads,
if the two portions of the prism be antecedently at-
tached to the bar by their centres. Since the whole
length of the bar is 2(w + «), the length of each half
is m -{- n. The distance of the point of suspension of
the right-hand portion of the prism from the point of
suspension of the bar is therefore m, and that of the
left-hand portion «. The experience that we have
here to deal with the weight, and not with the form,
of the bodies, is easily made. It is thus manifest, that
equilibrium will still subsist if any weight of the mag-
nitude 2m be suspended at the distance n on the one
side and any weight of the magnitude 2« be suspended
at the distance m on the other. The instinctive elements
of our perception of this phenomenon are even more

THE PRINCIPLES OF STATICS. 13

prominently displayed in this form of the deduction
than in that of Archimedes. .

We may discover, moreover, in this beautiful pre-
sentation, a remnant of the ponderousness which was
particularly characteristic of the investigators of an-
tiquity.

How a modem physicist conceived the same prob- Lagrange's
lem, may be learned from the following presentation of tion.
Lagrange. Lagrange says : Imagine a horizontal ho-
mogeneous prism suspended at its centre. Let this
prism (Fig. 7) be conceived divided into two prisms
of the lengths im and 2«. If now we consider the
centres of gravity of these two parts, at which we may
imagine weights to act proportional to 7.fn and iriy the

Fig. 7.

two centres thus considered will have the distances n
and m from the point of support. This concise dis-
posal of the problem is only possible to the practised
mathematical perception.

5. The object that Archimedes and his successors object of
sought to accomplish in the considerations we have here and his sac-
presented, consists in the endeavor to reduce the more
complicated case of the lever to the simpler and ap-
parently self-evident case, to discern the simpler in the
more complicated, or vice versa. In fact, we regard
a phenomenon as explained, when we discover in it
known simpler phenomena.

But surprising as the achievement of Archimedes
and his successors may at the first glance appear to
us, doubts as to the correctness of it, on further reflec-

14 THE SCIENCE OF MECHANICS.

Critique of tion, nevertheless spring up. From the mere assump-
ods. tion of the equilibrium of equal weights at equal dis-

tances is derived thp inverse proportionality of weight
and lever-arm ! How is that possible ? If we were
unable philosophically and a priori to excogitate the
simple fact of the dependence of equilibrium on weight
and distance, but were obliged to go for that result to
experience, in how much less a degree shall we be able,
by speculative methods, to discover the form of this
dependence, the proportionality !
The Statical As a matter of fact, the assumption that the equi-

moment in-
volved in librium- disturbing effect of a weight P 2X the distance

all their de- . ° . °

ductions. L from the axis of rotation is measured by the product
P,L (the so-called statical moment), is more or less
covertly or tacitly introduced by Archimedes and all
his successors. For when Archimedes substitutes for
a large weight a series of symmetrically arranged pairs
of small weights, which weights extend beyond the point
of support ^ he employs in this very act the doctrine of
the centre of gravity in its more general form, which is
itself nothing else than the doctrine of the lever in its
more general form.
Without it Without the assumption above mentioned of the im-
tion?°im* port of the product /'.Z, no one can prove (Fig. 8)

i>ossible. ^1 . * 1 1 1 •

that a bar, placed in
C^ any way on the ful-

crum Sy is supported,
with the help of a

I string attached to its

^^ centre of gravity and

^^^ ^' carried over a pulley,

by a weight equal to its own weight. But this is con-
tained in the deductions of Archimedes, Stevinus,
Galileo, and also in that of Lagrange.

THE PRINCIPLMS OF STATICS.

6. HuYGENS, indeed, reprehends this method, andHa»aiu'>
gives a different deduction, in which he believes he has
avoided the error. If in
the presentation of La-
grange we imagine the
two portions into which
the prism is divided
turned ninety degrees
about two vertical axes
passing through the cen-
tres of gravity s,s of the
prism -portions (see Fig.
911), and it be shown
that under these circum
stances equilibrium still .
continues to subsist, we Fig. 9.

shall obtain the Huygenian deduction. Abridged

and simplified, it is as follows.

ightless

plane (Fig. 9) through the p>oint S we draw a straight
line, on which we cut off on the one side the length i

THE SCIENCE OF MECHANICS.

His own de- and on the other the length 2, at A and B respectively.
On the extremities, at right angles to this straight
line, we place, with the centres as points of contact, the
heavy, thin, homogeneous prisms CD and EF^ of the
lengths and weights 4 and 2. Drawing the straight
line HSG (where AG=^\AC^ and, parallel to it, the
line CFy and translating the prism-portion CG by par-
allel displacement to FH^ everything about the axis
GH is symmetrical and equilibrium obtains. But
equilibrium also obtains for the axis AB ; obtains con-
sequently for every axis through S, and therefore also
for that at right angles to AB : wherewith the new
case of the lever is given.
Apparently Apparently, nothing else is assumed here than that
able. equal weights /,/ (Fig. 10) in the same plane and at

equal distances /,/ from an axis A A' (in this plane)
equilibrate one another. If we place ourselves in the
plane passing through A A* perpendicularly to /,/, say

Pig. zo.

Fig, II.

at the point M, and look now towards A and now
towards A\ we shall accord to this proposition the
same evidentness as to the first Archimedean proposi-
tion. The relation of things is, moreover, not altered if
we institute with the weights parallel displacements
with respect to the axis, as Huygens in fact does.

THE PRINCIPLES OF STATICS. 17

The error first arises in the inference : if equilib- Yet invow-
rium obtains for two axes of the plane, it also obtains final infer-

, - . • 1 1 1 . r • ®°ce an er-

for every other axis passing through the point of inter- ror.
section of the first two. This inference (if it is not to
be regarded as a purely instinctive one) can be drawn
only upon the condition that disturbant effects are as-
cribed to the weights proportional to their distances
from the axis. But in this is contained the very kernel
of the doctrine of the lever and the centre of gravity.
Let the heavy points of a plane be referred to a
system of rectangular co5rdinates (Fig. 11). The co-
ordinates of the centre of gravity of a system of masses
mm' m" , , , having the coSrdinates xx* x" , . . yy' y" . . .
are, as we know,

Mathemat-
^ 2mx 2my »cal discos-

inference.

If we turn the system through the angle a, the new co-
ordinates of the masses will be

x^ = x cosa — y sina, y^=y cosa -f- x sina

and consequently the coordinates of the centre of
gravity

J, Sm(xcosa — ysina) 2mx . 2my
S. = ^^ ^ = cosa -^ smar -.^^

= S cosa — Tf sina
and, similarly,

ff^ = ij cosa + S sina.

We accordingly obtain the coordinates of the new
centre of gravity, by simply transforming the coordi-
nates of the first centre to the new axes. The centre
of gravity remains therefore M^ self-same point. If
we select the centre of gravity itself as origin, then
2mx=^2my=i^. On turning the system of axes, this
relation continues to subsist. If, accordingly, equi-

i8 THE SCIENCE OF MECHANICS,

librium obtains for two axes of a plane that are per-
pendicular to each other, it also obtains, and obtains
then only, for every other axis through their point of
intersection. Hence, if equilibrium obtains for any
two axes of a plane, it will also obtain for every other
axis of the plane that passes through the point of in-
tersection of the two.
The infer- These conclusions, however, are not deducible if

enca admis-
sible only the coordinates of the centre of gravity are determined

on one con- .

dition. by some other, more general equation, say

tn -\- tn -\- tn -f- . . .

The Huygenian mode of inference, therefore, is in-
admissible, and contains the very same error that we
remarked in the case of Archimedes.
Seif-decep- Archimedes's self-deception in this his endeavor to
chimedes. reduce the complicated case of the lever to the case
instinctively grasped, probably consisted in his uncon-
scious employment of studies previously made on the
centre of gravity by the help of the very proposition he
sought to prove. It is characteristic, that he will not
trust on his own authority, perhaps even on that of
others, the easily presented observation of the import
of the product P.Ly but searches after a further verifi-
cation of it.

Now as a matter of fact we shall not, at least at
this stage of our progress, attain to any comprehension
whatever of the lever unless we directly discern in the
phenomena the product P,L as the factor decisive of
the disturbance of equilibrium. In so far as Archi-
medes, in his Grecian mania for demonstration, strives
to get around this, his deduction is defective. But re-
garding the import of P.L as given, the Archimedean

THE PRINCIPLES OF STATICS.

deductions still retain considerable value, in so far as Function of
the modes of conception of different cases are supported medean de-
the one on the other, in so far as it is shown that one
simple case contains all others, in so far as the same
mode of conception is established for all cases. Im-
agine (Fig. 12) a homogeneous prism, whose axis is
ABy supported at its centre C, To give a graphical
representation of the sum of the products of the weights
and distances, the sum decisive of the disturbance of
equilibrium, let us erect upon the elements of the axis,
which are proportional to the elements of the weight,
the distances as ordinates ; the ordinates to the right

Pig. IS.

of C*(as positive) being drawn upwards, and to the left illustration
of C (as negative) downwards. The sum of the areas ° *'* ^ "*
of the two triangles, A CD -f CBE = 0, illustrates here
the subsistence of equilibrium. If we divide the prism
into two parts at M, we may substitute the rectangle
MUWB for MTEB, and the rectangle MVXA for
TMCAD, where TP=\TE and TR = \TD, and the
prism-sections MB, MA are to be regarded as placed
at right angles to AB by rotation about Q and S.

THE SCIENCE OF MECHANICS,

In the direction here indicated the Archimedean
view certainly remained a serviceable one even after
no one longer entertained any doubt of the significance
of the product /'.Z, and after opinion on this point had
been established historically and by abundant verifica-
tion.
Treatment 7. The manner in which the laws of the lever, as

of the lever , , - ,

bjr modem handed down to us from Archimedes in their original
simple form, were further generalised and treated by
modern physicists, is very interesting and instructive.
Leonardo DA Vinci (1452-1519), the famous painter
and investigator, appears to have been the first to rec-
ognise the importance of the general notion of the so-

Fig. 13.

Leonardo called Statical moments. In the manuscripts he has
(X453-1519). left us, several passages are found from which this
clearly appears. He says, for example : We have a
bar AD (Fig. 13) free to rotate about A^ and suspended
from the bar a weight /*, and suspended from a string
which passes over a pulley a second weight Q. What
must be the ratio of the forces that equilibrium may ob-
tain? The lever- arm for the weight P is not ADy but
the "potential" lever AB, The lever-arm for the
weight ^ is not AD^ but the ** potential" lever AC,
The method by which Leonardo arrived at this view
is difficult to discover. But it is clear that he recog-

THE PRINCIPLES OF STATICS. 21

nised the essential circumstances by which the effect
of the weight is determined.

Considerations similar to those of Leonardo da Guide

Ubaldi.

Vinci are also found in the writings of Guido Ubaldi.
8. We will now endeavor to obtain some idea of
the way in which the notion of statical moment, by
which as we know is understood the product of a force
into the perpendicular let fall from the axis of rotation
upon the line of direction of the force, could have been
arrived at, — although the way that really led to this
idea is not now fully ascertainable. That equilibrium
exists (Fig. 14) if we lay a
cord, subjected at both sides
to equal tensions, over a
pulley, is perceived without
difficulty. We shall always
find a plane of symmetry for
the apparatus — the plane
which stands at right angles p'«- '♦•

to the plane of the cord and bisects {EE) the angle made
by its two parts. The motion that might be supposed a method
possible cannot in this case be precisely determined or the notion

, ^ , , , , . .,, , ofthestat-

defined by any rule whatsoever : no motion will there- icai mo-

, , , -r rill ment might

tore take place. If we note, now, further, that the mate- have been
rial of which the pulley ig made is only essential to the
extent of determining the form of motion of the points
of application of the strings,' we shall likewise easily
perceive that without disturbing the equilibrium of
the machine^ almost any portion of the pulley may be
removed. Essential remain only the rigid radii that
lead out to the tangential points of the string. We
see, thus, that the rigid radii (or the perpendiculars on
the linear directions of the strings) play here a part
similar to the lever-arms in the lever of Archimedes.

THE SCIENCE OF MECHANICS,

This notioD
derived
from the
considera-
tion of a
wheel and
axle.

Let US examine a so-called wheel and axle (Fig^.
15) of wheel-radius 2 and axle-radius i, provided re-
spectively with the cord-hung loads i and 2 ; an appa-
ratus which corresponds in every respect to the lever
of Archimedes. If now we place about the axle, in
any manner we may choose, a second cord, which we
subject at each side to the tension of a weight 2, the
second cord will not disturb the equilibrium. It is
plain, however, that we are also permitted to regard

Fig. 15. Fig. x6.

the two pulls marked in Fig. 16 as being in equilib-
rium, by leaving the two others, as mutually destruc-
tive, out of account. But we arrive in so doing, dis-
missing from consideration all unessential features, at
the perception that not only the pulls exerted by the
weights but also the perpendiculars let fall from the
axis on the lines of the pulls, are conditions deter-
minative of motion. The decisive factors are, then,
the products of the weights into the respective per-
pendiculars let fall from the axis on the directions of
the pulls ; ih other words, the so-called statical mo-
ments.
The princi- 9. What we have so far considered, is the devel-
fe Jar all* opmcnt of our knowledge of the principle of the lever,
explain the Quite independently of this was developed the knowl-

other in&.*

chines. edge of the principle of the inclined plane. It is not
necessary, however, for the comprehension of the ma-

THE PRINCIPLES OF STATICS.

chines, to search after a new principle beyond that of
the lever ; for the latter is sufficient by itself. Galileo,
for example, explains the inclined plane from the lever
in the following manner.
We have before us (Fig.
17) an inclined plane, on
which rests the weight
Qj held in equilibrium
by the weight F, Gali-
leo, now, points out the p>8- 17.
fact, that it is not requisite that Q should lie directly
upon the inclined plane, but that the essential point
is rather the form, or character, of the motion
of Q. We may, consequently, conceive the weight
attached to the bar AC, perpendicular to the inclined
plane, and rotatable about C If then we institute a Galileo's

. explanation

very slight rotation about the point C, the weight willofthein-
move in the element of an arc coincident with the in- plane by

the lever.

clined plane. That the path assumes a curve on the
motion being continued is of no consequence here,
since this further movement does not in the case of
equilibrium take place, and the movement of the in-
stant alone is decisive. Reverting, however, to the
observation before mentioned of Leonardo da Vinci,
we readily perceive the validity of the theorem Q, CB
= F. CA or Q/F =CA/CB = cafcb, and thus reach
the law of equilibrium on the inclined plane. Once we
have reached the principle of the lever, we may, then,
easily apply that principle to the comprehension of
the other machines.

TUB SCIENCE OF MECHANICS.

II.

Fig. i8.

THE PRINCIPLE OF THE INCLINED PLANE.

stevinus i. Stevinus, Of Stevin, (1548-1620) was the first

first investi- who iiivestie^ated the mechanical properties of the in-

gates the

mechftiiics clined plane : and he did so in an eminently original

of the in- r > . i_^,. /t-

clined , - manner. If a weight lie (Fig.

plane. / o v o

18) on a horizontal table, we
perceive at once, since the
pressure is directly perpendic-
ular to the plane of the table,
by the principle of symmetry,
that equilibrium subsists. On a
vertical wall, on the other hand, a weight is not at all
obstructed in its motion of descent. The inclined plane
accordingly will present an intermediate case between
these two limiting suppositions. Equilibrium will not
exist of itself, as it does on the horizontal support, but
it will be maintained by a less weight than that neces-
sary to preserve it on the vertical wall. The ascertain-
ment of the statical law that obtains in this case, caused
the earlier inquirers considerable difficulty.
His mode of Steviuus's manner of procedure is in substance as
law. follows. He imagines a triangular prism with horizon-

tally placed edges, a cross-section of which ABC is
represented in Fig. 19. For the sake of illustration
we will say that AB = iBC ; also that ^C is horizon-
tal. Over this prism Stevinus lays an endless string
on which 14 balls of equal weight are strung and tied
at equal distances apart. We can advantageously re
place this string by an endless uniform chain or cord.
The chain will either be in equilibrium or it will not.
If we assume the latter to be the case, the chain, since

THE PRINCIPLES OF STATICS. i;

the conditions of the event are not altered by its mo-
tion, must, when onc£ actually in motion, continue to
move for ever, that is, it must present a perpetual mo-
tion, which Stevinus deems absurd. Consequently only sieviai
the first case is conceivable. The chain remains in equi- ot ib"!
librium. The symmetrical portion y^iJC may, there- cUnad
fore, without disturbing the equilibrium, be removed. ''""'
The portion A£ of the chain consequently balances
the portion SC. Hence : on inclined planes of equal
heights equal weights act in the inverse proportion of
the lengths of the planes.

In the cross-section of the prism in Fig. 20 let us
imagine AC horizontal, .5 C vertical, and AB ^ 2BC;
furthermore, the chain-weights Q and P on AB and
BC proportional to the lengths ; it will follow then that

26 THE SCIENCE OF MECHANICS.

Q/P=i AB/BC=2, The generalisation is self-evi-
dent.
The as- 2. Unquestionably in the assumption from which

ofstevi"* Stevinus starts, that the endless chain does not move,
dncdon. there is contained primarily only a purely ins tine tw^
cognition. He feels at once, and we with him, that
we have never observed anything like a motion of the
kind referred to, that a thing of such a character does
not exist. This conviction has so much logical cogency
that we accept the conclusion drawn from it respecting
the law of equilibrium on the inclined plane without the
thought of an objection, although the law if presented
as the simple result of experiment, or otherwise put.
Their in- would appear dubious. We cannot be surprised at this
character, when We reflect that all results of experiment a^-e ob-
scured by adventitious circumstances (as friction, etc.),
and that every conjecture as to the conditions which are
determinative in a given case is liable to error. That
Stevinus ascribes to instinctive knowledge of this sort
a higher authority than to simple, manifest, direct ob-
servation might excite in us astonishment if we did not
ourselves possess the same inclination. The question
accordingly forces itself upon us : Whence does this
higher authority come ? If we remember that scientific
demonstration, and scientific criticism generally can
only have sprung from the consciousness of the individ-
ual fallibility of investigators, the explanation is not far
Their COR- to Seek. We feel clearly, that we ourselves have con-

ency.

tributed nothing to the creation of instinctive knowl-
edge, that we have added to it nothing arbitrarily, but
that it exists in absolute independence of our partici-
pation. Our mistrust of our own subjective interpre-
tation of the facts observed, is thus dissipated.

Stevinus's deduction is one of the rarest fossil in-

THE PRINCIPLES OF STATICS. 27

dications that we possess in the primitive history of HiRhhistor-
mechanics, and throws a wonderful light on the pro- stevinns's
cess of the formation of science generally, on its rise
from instinctive knowledge. We will recall to mind
that Archimedes pursued exactly the same tendency
as Stevinus, only with much less good fortune. In
later times, also, instinctive knowledge is very fre-
quently taken as the starting-point of investigations.
Every experimentator can daily observe in himself the
guidance that instinctive knowledge furnishes him. If
he succeed in abstractly formulating what is contained
in it, he will as a rule have made an important advance
in science.

Stevinus's procedure is no error. If an error were The trust-
contained in it, we should all share it. Indeed, it is of instinc-
perfectly certain, that the union of the strongest in- edge,
stinct with the greatest power of abstract formulation
alone constitutes the great natural inquirer. This by
no means compels us, however, to create a new mysti-
cism out of the instinctive in science and to regard this
factor as infallible. That it is not infallible, we very
easily discover. Even instinctive knowledge of so
great logical force as the principle of symmetry em-
ployed by Archimedes, may lead us astray. Many of
my readers will recall to mind, perhaps, the intellectual
shock they experienced when they heard for the first
time that a magnetic needle lying in the magnetic
meridian is deflected in a definite direction away from
the meridian by a wire conducting a current being car-
ried along in a parallel direction above it. The instinc-
tive is just as fallible as the distinctly conscious. Its only
value is in provinces with which we are very familiar.
Let us rather put to ourselves, in preference to
pursuing mystical speculations on this subject, the

28 THE SCIENCE OF MECHANICS.

The origin question : How does instinctive knowledge originate

of instinc-

tiveknowi- and what are its contents? Everything which we ob-
serve in nature imprints itself uncomprehended and u^ti-
analysed in our percepts and ideas, which, then, in their
turn, mimic the processes of nature in their most gen-
eral and most striking features. In these accumulated
experiences we possess a treasure-store which is ever
close at hand and of which only the smallest portion
is embodied in clear articulate thought. The circum-
stance that we are easier able to employ these expe-
riences than we are nature itself, and that they are,
notwithstanding this, free, in the sense indicated, from
all subjectivity, invests them with a high value. It
is a peculiar property of instinctive knowledge that it
is predominantly of a negative nature. We cannot so
well say what must happen as we can what cannot hap-
pen, since the latter alone stands in glaring contrast to
the obscure mass of experience in us in which single
characters are not distinguished.
Instinctive Still, great as the importance of instinctive knowl-
and%tera-edge may be, for discovery, we must not, from our
mutnaiiy poiut of vicw, rest Content with the recognition of its
each other, authority. We must inquire, on the contrary : Under
what conditions could the instinctive knowledge in
question have originated? We then ordinarily find that
the very principle to establish which we had recourse
to instinctive knowledge, constitutes in its turn the fun-
damental condition of the origin of that knowledge.
And this is quite obvious and natural. Our instinctive
knowledge leads us to the principle which explains that
knowledge itself, and which is in its turn also corrobo-
rated by the existence of that knowledge, which is a
separate fact by itself. This we will find on close ex-
amination is the state of things in Stevinus's case.

THE PRINCIPLES OF ST A TICS, 29

3. The reasoning of Stevinus impresses us as soTheinRen-
highly ingenious because the result at which he arrives vinns's rea-
apparently contains more than the assumption from
which he starts. While on the one hand, to avoid con-
tradictions, we are constrained to let the result pass, on
the other an incentive remains which impels us to seek
further insight. If Stevinus had distinctly set forth
the entire fact in all its aspects, as Galileo subsequently
did, his reasoning would no longer strike us as ingen-
ious ; but we should have obtained a much more satis-
factory and clear insight into the matter. In the
endless chain which does not glide upon the prism, is
contained, in fact, everything. We might say, the
chain does not glide because no sinking of heavy bodies
takes place here. This would not be accurate, how-
ever, for when the chain moves many of its links really
do descend, while others rise in their place. We must
say, therefore, more accurately, the chain does not
glide because for every body that could possibly de- critique of
scend an equally heavy body would have to ascend deduction,
equally high, or a body of double the weight half the
height, and so on. This fact was familiar to Stevinus,
who presented it, indeed, in his theory of pulleys ;
but he was plainly too distrustful of himself to lay
down the law, without additional support, as also valid
for the inclined plane. But if such a law did not exist
universally, our instinctive knowledge respecting the
endless chain could never have originated. With this
our minds are completely enlightened. — The fact that
Stevinus did not go as far as this in his reasoning and
rested content with bringing his (indirectly discovered)
ideas into agreement with his instinctive thought, need
not further disturb us.

The service which Stevinus renders himself and his

30 THE SCIENCE OF MECHANICS.

hemsrit readers, consists, therefore, in the contrast and coin-

ji-aproce- parison of knowledge that is instinctive with knowledge

that is clear, in the bringing the two into connection

and accord with one another, and in the supporting

the one upon the other. The strengthening of mental
view which Stevinus acquired by this procedure, we
learn from the fact that a picture of the endless chain
and the prism graces as vignette, with the inscrlptioa
"Wonder en is gheen wonder," the title-page of his

THE PRINCIPLES OF STATICS.

work Hypomnemata Mathematica (Leyden, 1605).* As
a fact, every enlightening progress made in science is
accompanied with a certain feeling of disillusionment.
We discover that that which appeared wonderful to
us is no more wonderful than other things which we
know instinctively and regard as self-evident ; nay,
that the contrary would be much more wonderful ; that
everywhere the same fact expresses itself. Our puzzle
turns out then to be a puzzle no more ; it vanishes into
nothingness, and takes its place among the shadows
of history.

4. After he had arrived at the principle of the in-
clined plane, it was easy for Stevinus to apply that
principle to the other machines; and to explain by it
their action. ^He makes, for example, the following
application. ■

We have, let us suppose, an inclined plane (Fig.
22) and on it a load Q, We pass a string over the
pulley A at the summit and imagine the load Q held in
equilibrium by the load P,
Stevinus, now, proceeds by
a method similar to that
later taken by Galileo. He
remarks that it is not ne-
cessary that the load Q
should lie directly on the
inclined plane. Provided ' p»k- ««

only the form of the machine's motion be preserved, the
proportion between force and load will in all cases re-
main the same. We may therefore equally well conceive
the load Q to be attached to a properly weighted string
passing over a pulley D: which string is normal to the

*The title given is that of Willebrord Snell's Latin translation (1608) of
Simon Stevin's IVisconstigt Gtdackttnisstnt Leyden, i6o5.~7Va»r.

Enlighten-
ment in
science al-
ways ac-
companied
with disillu-
sionment.

Explana-
tion of the
other ma-
chines by
Stevinus's
principle.

THE SCIENCE OF MECHANICS,

The funicu-
lar machine

And the
special case
ox the paral-
lelogram of
forces.

The general
form of the
last-men-
tioned prin-
ciple also
employed.

inclined plane. If we carry out this alteration, ive
shall have a so-called funicular machine. We nomr
perceive that we can ascertain very easily the portion
of weight with which the body on the inclined plane
tends downwards. We have only to draw a vertical
line and to cut off on it a portion ab corresponding to
the load Q, Then drawing on aA the perpendicular
bcy we have F/Q = AC/AB = ac/ab. Therefore ac
represents the tension of the string aA, Nothing pre-
vents us, now, from making the two strings change
functions and from imagining the load Q to lie on the
dotted inclined plane EDF. Similarly, here, we ob-
tain ad for the tension of the second string. In this
manner, accordingly, Stevinus indirectly arrives at a
knowledge of the statical relations of the funicular
machine and of the so-called parallelogram of forces ; at
first, of course, only for the particular case of strings
(or forces) ac^ ad at right angles to one another.

Subsequently, indeed, Stevinus employs the prin-
ciple of the composition and resolution of forces in
a more general form ; yet the method by which he

Fig. 23 Fi(5. 34.

reached the principle, is not very clear, or at least is
not obvious. He remarks, for example, that if we
have three strings AB^ AC, AD, stretched at any

THE PRINCIPLES OF STATICS.

given angles, and the weight F is suspended from the
first, the tensions may be determined in the following
manner. We produce (Fig. 23) AB to X and cut off
on it a portion AE. Drawing from the point E^ EF
parallel to AD and EG paral-
lel to A C\ the tensions of AB,
AC, AD are respectively pro-
portional to AEj AF, AG.

With the assistance of this
principle of construction Ste-
vinus solves highly compli- fir. 25.

Gated problems. He determines, for instance, the solution of
tensions of a system of ramifying strings like that pUcYted"^
illustrated in Fig. 24; in doing which of course he^'** **™*'
starts from the given tension of the vertical string.

The relations of the tensions of a funicular polygon
are likewise ascertained by construction, in the man-
ner indicated in Fig. 25.

We may therefore, by means of the principle of the General re-
inclined plane, seek to elucidate the conditions of op-
eration of the other simple machines, in a manner sim-
ilar to that which we employed in the case of the prin-
ciple of the lever.

III.

THE PRINCIPLE OF THE COMPOSITION OF FORCES.

I. The principle of the parallelogram of forces, atTheprinci-

which Stevinus arrived and employed, (yet without ex- parallelog-
ram of

pressly formulating it,) consists, as we know, of thefoi
following truth. If a body A (Fig. 26) is acted upon
by two forces whose directions coincide with the lines
AB and AC, and whose magnitudes are proportional to
the lengths AB and AC, these two forces produce the

orces.

THE SCIENCE OF MECHANICS,

Pig. 26.

same effect as a single force, which acts in the direction
of the diagonal AD of the parallelogram ABCD and is
proportional to that diagonal. For instance, if on the

strings AB^ AC weights
exactly proportional to the
lengths AB^ AC be sup-
posed to act, a single
weight acting on the string
>4Z> exactly proportional to
the length AD will produce the same effect as the first
two. The forces AB and A C are called the compo-
nents, the force AD the resultant. It is furthermore
obvious, that conversely, a single force is replaceable
by two or several other forces.
Method by 2. We shall now endeavor, in connection with the

which the

general no- investigations of Stevinus, to give ourselves some idea

tion of the » o

paraiieio- of the manner in which the

gram of \,^ Jf

forces \ f general proposition of the

beenar- \^^^^-"^"'|\ parallelogram of forces

rived at. Af^^ si \ • i. , l • j

might have been arrived
at. The relation, — dis-
covered by Stevinus, —
that exists between two
mutually perpendicular
forces and a third force
that equilibrates them, we
shall assume as (indi-
rectly) given. We sup-
pose now (Fig. 27) that
there act on three strings
OX, O y, OZ, pulls which

Fig. 27.

balance each other. Let us endeavor to determine the
nature of these pulls. Each pull holds the two remain-
ing ones in equilibrium. The pull (7Kwe will replace

THE PRINCIPLES OF STATICS. 35

(following Stevinus's principle) by two new rectangular The dedac-
pulls, one in the direction Ou (the prolongation ofRenerai

principie

OX\ and one at right angles thereto in the direction xrom the

"^ c» t» ^ special case

Ov, And let us similarly resolve the pull OZ in theofstevinus.
directions Ou and Ow. The sum of the pulls in the di-
rection Ouy then, must balance the pull OX, and the
two pulls in the directions Ov and Oiv must mutually
destroy each other. Taking the two latter as equal
and opposite, and representing them by Om and On,
we determine coincidently with the operation the com-
ponents Op and Of parallel to Ou, as well also as the
pulls Or, Os, Now the sum Op -j- Oq is equal and op-
posite to the pull in the direction of OX ; and if we
draw st parallel to O Y, or rt parallel to OZ, either line
will cut off the portion Ot =^ Op -\- Oq : with which re-
sult the general principle of the parallelogram of forces
is reached.

The general case of composition may be deduced a different
in still another way from the special composition of same de-
rectangular forces. Let OA and OB be the two forces
acting at ^. For ^.5 substitute
a force OC acting parallel to
OA and a force OD acting at
right angles to OA, There
then act for OA and OB the
two forces OE = OA -\- OC ^^' ^•

and OD, the resultant of which forces OF is at the same
time the diagonal of the parallelogram OAFB con-
structed on OA and OB as sides.

^ 3. The principle of the parallelogram of forces, The pnn-
when reached by the method of Stevinus, presents it- presena it-
self as an indirect discovery. It is exhibited as a con- indirect
sequence and as the condition of known facts. We
perceive, however, merely that it does exist, not, as yet

0 (

C A E

■^\

i J

5 7-

THE SCIENCE OF MECHANICS.

And is first why it exists ; that is, we cannot reduce it (as in dy-
enunciated namics) to Still simpler propositions. In statics, in-
and varig- deed, the principle was not fully admitted until the
time of Varignon, when dynamics, which leads directly
to the principle, was already so far advanced that its
adoption therefrom presented no difficulties. The prin-
ciple of the parallelogram of forces was first clearly
^enunciated by Newton in his Principles of Natural Phi-
losophy. In the same year, Varignon, independently of
Newton, also enunciated the principle, in a work sub-
mitted to the Paris Academy (but not published un-
til after its author's death), and made, by the aid of a
geometrical theorem, extended practical application
of it.*

The geometrical theorem referred to is this. If we
consider (Fig. 29) a parallelogram the sides of which
are/ and ^, and the diagonal is r, and from any point m

in the planq of the par-
allelogram we draw per-
pendiculars on these
three straight lines,
which perpendiculars
we will designate as
Uy V, Wy then /.«-(-
q , V :=. r . w. This is
easily proved by draw-
Fig. 39. Fig. 30. ing straight lines from m
to the extremities of the diagonal and of the sides of
the parallelogram, and considering the areas of the
triangles thus formed, which are equal to the halves
of the products specified. If the point m be taken
within the parallelogram and perpendiculars then be

The geo-
metrical
theorem
emplojred
by Varig-
non.

* In the same year, 1687, Father Bernard Lami published a little appendix
to his Traiti <U michanique^ developing the same principle.— TVtfjw.

THE PRINCIPLES OF STATICS. yj

drawn, the theorem passes into the form p ,u — q ,v
z=r .w. Finally, if m be taken on the diagonal and
perpendiculars again be drawn, we shall get, since the
perpendicular let fall on the diagonal is now zero,
/. u — ^.z; = 0 or/, u ^= q ,v.

With the assistance of the observation that forces The dedoc-
are proportional to the motions produced by them in
equal intervals of time, Varignon easily advances from
the composition of motions to the composition of forces.
Forces, which acting at a point are represented in
magnitude and direction by the sides of a parallelo-
gram, are replaceable by a single force, similarly rep-
resented by the diagonal of that parallelogram.

If now, in the parallelogram considered,/ and ^Moments of
represent the concurrent forces (the components) and r °'^"'*
the force competent to take their place (the resultant),
then the products pu, qv, rw are called the moments
of these forces with respect to the point m. If the point
m lie in the direction of the resultant, the two moments
pu and qv are with respect to it equal to each other.

4. With the assistance of this principle Varignon is vangnon's

•-.• . . - treatment

now m a position to treat y of the sim-

the machines in a much \ / ?h1nes.

simpler manner than were
his predecessors. Let us
consider, for example,
(Fig. 31) a rigid body
capable of rotation about
an axis passing through
O, Perpendicular to the
axis we conceive a plane,
'and select therein two ^'^ ^''

points Ay B, on which two forces P and Q in the plane
are supposed to act. We recognise with Varignon

38 THE SCIENCE OF MECHANICS.

Tho deduc- that the effect of the forces is not altered if their points

ttnn of the

law of the of application be displaced along their line of action,

lever from . , .

the parai- since all points in the same direction are rigidly con-

lelosram*

principle, nected with one another and each one presses and pulls
the other. We may, accordingly, suppose P applied
at any point in the direction AX, and Q at any point
in the direction BY^ consequently also at their point
of intersection M. With the forces as displaced to My
then, we construct a parallelogram, and replace the
forces by their resultant. We have now to do only
with the effect of the latter. If it act only on movable
points, equilibrium will not obtain. If, however, the
direction of its action pass through the axis, through
the point O, which is not movable, no motion can take
place and equilibrium will obtain. In the latter case
^ is a point on the resultant, and if we drop the per-
pendiculars u and V from O on the directions of the
forces /, q, we shall have, in conformity with the the-
orem before mentioned, p - u=zq • v. With this we
have deduced the law of the lever from the principle
of the parallelogram of forces.
The statics Varignon explains in like manner a number of other
adyna2?csa cases of equilibrium by the equilibration of the result-
ant force by some obstacle or restraint. On the in-
clined plane, for example, equilibrium exists if the re-
sultant is found to be at right angles to the plane. In
fact, Varignon rests statics in its entirety on a dynamic
foundation ; to his mind, it is but a special case of dy-
namics. The more general dynamical case constantly
hovers before him and he restricts himself in his inves-
tigation voluntarily to the case of equilibrium. We
are confronted here with a dynamical statics, such
as was possible only after the researches of Galileo.
' Incidentally, it may be remarked, that from Varignon

THE PRINCIPLES OF STA TICS,

is derived the majority of the theorems and methods
of presentation which make up the statics of modem
elementary text-books.

5. As we have abready seen, purely statical consid- special

. . staticalcoo-

erations also lead to the proposition of the parallel- siderations
ogram of forces. In special cases, in fact, the principle the prin-
admits of being very easily verified. We recognise at
once, for instance, that any number whatsoever of equal
forces acting (by pull or pressure) in the same plane at
a point,' around which their suc-
cessive lines make equal angles,
are in equilibrium. If, for exam-
ple, (Fig. 32) the three equal
forces OAj OB, OC act on the
point O at angles of 120°, each
two of the forces holds the third
in equilibrium. We see imme-
diately that the resultant of OA
and OB is equal and opposite to ^C It is represented
by OD and is at the same time the diagonal of the
parallelogram OADB, which readily follows from the
fact that the radius of a circle is also the side of the
hexagon included by it.

6. If the concurrent forces act in the same or in The case of
opposite directions, the resultant is equal to the sum forces ^
or the difference of the parttcaiar

_ . t«r O A case of the

components. We rec- 1 — ^^-^ ^ Renerai

ognise both cases with- B* ^ r» C P^ncipie.

out any difficulty as
particular cases of the
principle of the paral-
lelogram of forces. If in the two drawings of Fig. 33
we imagine the angle A OB to be gradually reduced
to the value 0°, and the angle A' ff B' increased to the

Fig. 3a.

Fig. 33.

40 THE SCIENCE OF MECHANICS,

value 1 80°, we shall perceive that repasses into OA -(-
AC=OA-^OBzxi^ a C into a A — A C = a A*
— (y B. The principle of the parallelogram of forces
includes, accordingly, propositions which are generally
made to precede it as independent theorems.
The princi- 7. The principle of the parallelogram of forces, in

pteapropo- ***«-'

shionde- the form in which it was set forth by Newton and

rived from , , , .

eiperieuce. Varignon, clearly discloses itself as a proposition de-
rived from experience. A point acted on by two forces
describes with accelerations proportional to the forces
two mutually independent motions. On this fact the
parallelogram construction is based. Daniel Ber-
noulli, however, was of opinion that the proposition of
the parallelogram of forces was a geometrical truth, in-
dependent of physical experience. And he attempted
to furnish for it a geometrical demonstration, the chief
features of which we shall here take into consideration,
as the Bernoullian view has not, even at the present
day, entirely disappeared.
Daniel Ber- If two equal forces, at right angles to each other
tempted (Fig. 34), act on a point, there can be no doubt, ac-
Seuons^a- ^ cording to Bernoulli, that the line

^ y1\ o^ bisection of the angle (con-

formably to the principle of sym-
metry) is the direction of the re-
sultant r. To determine geomet-
rically also the magnitude of the
resultant, each of the forces / is
^**- 3^' decomposed into two equal forces

qy parallel and perpendicular to r. The relation in
respect of magnitude thus produced between / and q
is consequently the same as that between r and /. We
have, accordingly:

p = ^, q and r =1 fx . f\ whence r = fx'^q.

fometricai
euonstra

tion of the

truth

THE PRINCIPLES OF STATICS.

Since, however, the forces q acting at right angles
to r destroy each other, while those parallel to r con-
stitute the resultant, it further follows that

r = 2^; hence jx = l/2, and r = i/2 . /.

The resultant, therefore, is represented also in re-
spect of magnitude by the diagonal of the square con-
structed on / as side.

Similarly, the magnitude may be determined of the The case of
resultant of unequal rectangular components. Here, rectangaiar

however, nothing is known before- ^

hand concerning the direction of
the resultant r. If we decompose
the components /, q (Fig. 35), A
parallel and perpendicular to the
yet undetermined direction r, into
the forces «, j and v, /, the new
forces will form with the compo-
nents /, q the same angles that fi,
q form with r. From which fact the following relations
in respect of magnitude are determined :

Fig. 35.

u a V

s p t'

p u Q '^ Q

from which two latter equations follows s =z t = pq/r.
On the other hand, however.

r = «+» = -"—+ -i^ or r* =/» + g*.

r r

The diagonal of the rectangle constructed on / and
q represents accordingly the magnitude of the result-
ant.

Therefore, for all rhombs, the direction of the re- General re-
sultant is determined ; for all rectangles, the magni-
tude; and for squares both magnitude and direction.
Bernoulli then solves the problem of substituting for

42 THE SCIENCE OF MECHANICS,

two equal forces acting at one given angle, other equal,
equivalent forces acting at a different angle ; and finally
arrives by circumstantial considerations, not wholly
exempt from mathematical objections, but amended
later by Poisson, at the general principle.
Critiqne of 8. Let US uow examine the physical aspect of this

Bernoulli's '^' a^ A t

method, question. As a proposition derived from experience,
the principle of the parallelogram of forces was already
known to Bernoulli. What Bernoulli really does, there-
fore, is to simulate towards himself a complete ignorance
of the proposition and then attempt to philosophise
it abstractly out of the fewest possible assumptions.
Such work is by no means devoid of meaning and pur-
pose. On the contrary, we discover by such proce-
dures, how few and how imperceptible the experiences
are that suffice to supply a principle. Only we must
not deceive ourselves, as Bernoulli did ; we must keep
before our minds all the assumptions, and should over-
look no experience which we involuntarily employ.
What are the assumptions, then, contained in Bernoul-
li's deduction?
The as- Q. Statics, primarily, is acquainted with force only

ofhlsd"* as a pull or a pressure, that from whatever source it
rived^from may come always admits of being replaced by the pull
experience. ^^ ^^ pressure of a weight. All forces thus may be re-
garded as quantities of the same kind and be measured
by weights. Experience further instructs us, that the
particular factor of a force which is determinative of
equilibrium or determinative of motion, is contained
not only in the magnitude of the force but also in its
direction^ which is made known by the direction of the
resulting motion, by the direction of a stretched cord,
or in some like manner. We may ascribe magnitude
indeed to other things given in physical experience,

THE PRINCIPLES OE STATICS. 43

such as temperature, potential function, but not direc-
tion. The fact that both magnitude and direction are
determinative in the efficiency of a force impressed on
a point is an important though it may be an unob-
trusive experience.

Granting, then, that the magnitude and direction M^itude
of forces impressed on a point alone are decisive, it will tion the sole
be perceived that two equal and opposite forces, as they factors,
cannot uniquely and precisely determine any motion,
are in equilibrium. So, also, at
right angles to its direction, a
force/ is unable uniquely to de- '

termine a motional effect. But

if a force / is inclined at an an-
gle to another direction ss* (Fig.

36), it is able to determine a mo-

tion in that direction. Yet ex- ^' *

perience alone can inform us, **' ^ *

that the motion is determined in the direction of s* s
and not in that of x/ ; that is to say, in the direction
of the side of the acute angle or in the direction of the
projection of / on s's.

Now this latter experience is made use of by Ber- Thec^ee/of
noulli at the very start. The sense, namely, of the re- derfSabS
sultant of two equal forces acting at right angles to one exiMrienca
another is obtainable only on the ground of this expe-
rience. From the principle of symmetry follows only,
that the resultant falls in the plane of the forces and
coincides with the line of bisection of the angle, not
however that it falls in the acute angle. But if we sur-
render this latter determination, our whole proof is ex-
ploded before it is begun.

10. If, now, we have reached the conviction that
our knowledge of the effect of the direction of a force is

44 THE SCIENCE OF MECHANICS.

So also solely obtainable from experience, still less then shall

must the , ,- -^ . . . , .,

form of the we believe it m our power to ascertain by any other way

efftfct be

thusde- the^r^i of this effect. It is utterly out of our power,
to divine, that a force p acts in a direction s that makes
with its own direction the angle or, exactly as a force
/ cos a in the direction s ; a statement equivalent to the
proposition of the parallelogram of forces. Nor was
it in Bernoulli's power to do this. Nevertheless, he
makes use, scarcely perceptible it is true, of expe-
riences that involve by implication this very mathe-
matical fact.
The man- A person already familiar with the composition

which the and resolution of forces is well aware that several forces

assump-
tions men- acting at a point are, as regards their effect, replaceable,

tioned enter , - . . -

into Bcr- m evcry respect and in every direction, by a single force.

ducUon. This knowledge, in Bernoulli's mode of proof, is ex-
pressed in the fact that the forces />, q are regarded as
absolutely qualified to replace in all respects the forces
J, u and /, v, as well in the direction of r as in every
other direction. Similarly r is regarded as the equiv-
alent of / and q. It is further assumed as wholly in-
different, whether we estimate j, «, /, v first in the
directions of/, q^ and then/, q in the direction of r, or
J, «, /, v be estimated directly and from the outset in
the direction of r. But this is something that a person
only can know who has antecedently acquired a very
extensive experience concerning the composition and
resolution of forces. We reach most simply the knowl-
edge of the fact referred to, by starting from the knowl-
edge of another fact, namely that a force / acts in a
direction making with its own an angle a^ with an effect
equivalent to p • cos a. As a fact, this is the way the
perception of the truth was reached.

Let the coplanar forces P, P\ P*\ . . be applied to

THE PRINCIPLES OF STA TICS. 45

one and the same point at the angles a, a\ or" . . . with Mathemat-

leal analy~

a given direction X. These forces, let us suppose, are sis of the
replaceable by a single force U, which makes with Jf the true and

necessary

an angle /i. By the familiar principle we have then assumptioa

2Pcosa = 7Icos/i.

If n is still to remain the substitute of this system of
forces, whatever direction X may take on the system
being turned through any angle 6, we shall further
have

:^/'cos (a + tf) == 77 COS (pi + 6),
or

(^Pcosa—IIcospi) cos6 — {2Psina — ilsin/^) sind = 0.

If we put

2P cosa — 77 cos// = A,

— {2Psina — 77 sin//) == B,

B
tanr = j-,
A

it follows that

A cos6 + B sintf = \/A^ + B^ sin (rf + t) = 0,

which equation can subsist for ^vrry 6 only on the con-
dition that

A = 2Pcosa — IJcosju = 0
and

B = {2Fsina — 11 sin//) = 0 ;

whence results

77 cos// = 2P cosa

77 sin// = 2Psina.

From these equations follow for 77 and // the deter-
minate values *

n=i/l{2Psmay + (2Pcosay'\

and

2Psina > '

tan// = ^^ .

2 P cosa

46 THE SCIENCE OF AfECHANICS,

The actual Granting, therefore, that the effect of a force in every

resnlts not _.. , j,. •• tt*

deducibie direction Can be measured by its projection on that di-

on any • i i i. #•

other sup- rection, then truly every system of forces acting at a
point is replaceable by a single force, determinate in
magnitude and direction. This reasoning does not hold,
however, if we put in the place of cos a any general func-
tion of an angle, (p (a). Yet if this be done, and we still
regard the resultant as determinate, we shall obtain for
9? (a), as may be seen, for example, from Poisson's
deduction, the form cos a. The experience that several
forces acting at a point are always, in every respect,
replaceable by a single force, is therefore mathemat-
ically equivalent to the principle of the parallelogram
of forces or to the principle of projection. The prin-
ciple of the parallelogram or of projection is, how-
ever, much easier reached by observation than the

General re- more general experience above mentioned by statical
observations. And as a fact, the principle of the par-
allelogram was reached earlier. It would require in-
deed an almost superhuman power of perception to
deduce mathematically, without the guidance of any
further knowledge of the actual conditions of the ques-
tion, the principle of the parallelogram from the gen-
eral principle of the equivalence of several forces to a
single one. We criticise accordingly in the deduction
of Bernoulli this, that that which is easier to observe
is reduced to that which is more difficult to observe.
This is a violation of the economy of science. Bernoulli
is also deceived in imagining that he does not proceed
from any fact whatever of observation.

An addi- We must further remark that the fact that the forces

aumption of are independent of one another, which is involved in

Bernoulli. ., i r ^i • '^^ • .^i

the law of their composition, is another experience
which Bernoulli throughout tacitly employs. As long

THE PRINCIPLES OF STATICS. 47

as we have to do with uniform or symmetrical systems
of forces, all equal in magnitude, each can be affected
by the others, even if they are not independent, only
to the same extent and in the same way. Given but
three forces, however, of which two are symmetrical
to the third, and even then the reasoning, provided
we admit that the forces may not be independent, pre-
sents considerable difticulties.

11. Once we have been led, directly or indirectly. Discussion
to the principle of the parallelogram of forces, once we scter of Oie
have perceived it, the principle is just as much an ob-** ^**
servation as any other. If the observation is recent, it

of course is not accepted with the same confidence as
old and frequently verified observations. We then seek
to support the new observation by the old, to demon-
strate their agreement. By and by the new observa-
tion acquires equal standing with the old. It is then
no longer necessary constantly to reduce it to the lat-
ter. Deduction of this character is expedient only in
cases in which observations that are difficult directly
to obtain can be reduced to simpler ones more easily
obtained, as is done with the principle of the parallel-
ogram of forces in d3mamics.

12. The proposition of the parallelogram of forces Bzi>erimen-
has also been illustrated by experiments especially tfon of' he
instituted for the purpose. An apparatus very wellScontnv-
adapted to this end was contrived by Cauchy. The Suchy.
centre of a horizontal divided circle (Fig. 37) is marked

by a pin. Three threads /,/*, /", tied together at a
point, are passed over grooved wheels r, r*, r", which
can be fixed at any point in the circumference of the
circle, and are loaded by the weights /, /', /". If three
equal weights be attached, for instance, and the wheels
placed at the marks of division o, 120, 240, the point at

THE SCIENCE OF MECHANICS.

Eipsrimea- which the Strings are knotted will assume a position
1 just above the centre of the circle. Three equal forces
acting at angles of 120°, accordingly, are in equilib-
rium.

principle.

If we wish to represent another and different case,
we may proceed as follows. We imagine any two
forces/, g acting at any angle a, represent (Fig. 38)
them by lines, and construct on them as sides a paral-
lelogram. We supply, further, a force
equal and opposite to the resultant r.
The three forces p,q, — r hold each
other in equilibrium, at the angles vis-
ible from the construction. We now
place the wheels of the divided circle on
the points of division o, a, a-\- j5, and
load the appropriate strings with the
weights/, q, r. The point at which the
strings are knotted will come to a position exactly
above the middle point pf thq circle.

n«-38.

THE PRINCIPLES OF STA TICS, 49

IV.
THE PRINCIPLE OF VIRTUAL VELOCITIES.

I . We now pass to the discussion of the principle The truth

. . of the prin-

of virtual (possible) displacements.* The truth of cipie first

^ ' remarked

this principle was first remarked by Stevinus at the by stevinus
close of the sixteenth century in his investigations on
the equilibrium of pulleys and combinations of pulleys.
Stevinus treats combinations of pulleys in the same
way they are treated at the present day. In the case

* Termed in English the principle of " virtual velocities," this being the
original phrase (vUesst virtuelU) introduced by John Bernoulli. See the
text, page 56. The word virtualis seems to have been the fabrication of Duns
Scotus (see the Century Dictionary ^ under virtual) ; but virtualiter was used
by Aquinas, and virtu* had been employed for centuries to translate dbvafUC,
and therefore as a synonym for potentia. Along with many other scholastic
terms, virtual passed into the ordinary vocabulary of the English language.
Everybody remembers the passage in the third book of Paradise Lost^

" Love not the heav'niy Spirits, and how thir Love
Express they, by looks onely, or do they mix
Irradiance, virttiai or immediate touch r " — Milton.

So, we all remember how it was claimed before our revolution that America
had " p/r/aM/ representation " in parliament. In these passages, as in Latin,
virtual means : existing in effect, but not actually. In the same sense, the
word passed into French ; and was made pretty common among philosophers
by Leibnitz. Thus, he calls innate ideas in the mind of a child, not yet brought
to consciousness, "des connoissances virtuelles.*^ This does not mean " pos-
sible," but just what v/r/wa/ ordinarily means now, as just defiuQd.

The principle in question was an extension to the case of more than two
forces of the old rule that "what a machine gains in power ^ it loses in velocity.''*
Bernoulli's modification reads that the sum of the products of the forces intojl
their virtual velocities must vanish to give equilibrium. He says, in effect : ||
give the system any possible and infinitesimal motion you please, and then
the simultaneous displacements of the points of application of the forces,
resolved in the directions (^ those for ces^ though they are not exactly velocities,
since they are only displacements in one time, are, nevertheless, virtually
velocities, for the purpose of applying the rule that what a machine gains in
power, it loses in velocity.

Thomson and Tait say : " If the point of application of a force be dis-
placed through a small space, the resolved part of the displacement in the di-
rection of the force has been called its Virtual Velocity. This is positive or
negative according as the virtual velocity is in the same, or in the opposite,
direction to that of the force." This agrees with Bernoulli's definition which
may be found in Varignon's Nouvelle micanique^ Vol. II, Chap, vl.— Trans,

so THE SCIENCE OF MECHANICS.

sie^noi'« a (Fig. 39) equilibrium obtains, when an equal weight P
''on« ™ ihe is suspended at each aide, for reasons already familiar.
ofpuiiejB. In b, the weight P is suspended by two parallel cords.

each of which accordingly supports the weight /*/2,
with which weight in the case of equilibrium the free
end of the cord must also be loaded. In c, P is sus-
pended by six cords, and the weighting of the free ex-
tremity with /'/6 will accordingly produce equilibrium.
\Tid, the so-called Archimedean or potential pulley,* P
in the first instance is suspended by two cords, each
of which supports Pji ; one of these two cords in turn
is suspended by two others, and so on to the end, so
that the free extremity will be held in equilibrium by
the weight Pj%. If we impart to these assemblages
of pulleys displacements corresponding to a descent of
the weight P through the distance h, we shall observe
that as a result of the arrangement of the cords

the counterweight P

a distance A i

Pl^

p/e

will ascend

P/8

" " 8A '

THE PRINCIPLES OF STATICS. 51

In a system of pulleys in equilibrium, therefore, His conciu-

' 810ns too

the products of the weights into the displacements germ of the
they sustain are respectively equal. (** Ut spatium
agentis ad spatium patientis, sic potentia patientis ad
potentiam agentis." — Stevini, Hypomnematay T. IV,
lib. 3, p. 172.) In this remark is contained the germ
of the principle of virtual displacements.

2. Galileo recognised the truth of the principle in GaiUeo**

, , , , . , recognition

another case, and that a somewhat more general one ; of the prin-

• . ciple in the

namely, in its application to the mclined plane. On case of the
an inclined plane (Fig. 40), plane,

the length of which AB is
double the height BC, 3. load
Q placed on AB is held in
equilibrium by the load P act-
ing along the height BCy if
P = Q/i, If the machine be

set in motion, P = Q/2 will descend, say, the vertical
distance h, and Q will ascend the same distance h along
the incline AB, Galileo, now, allowing the phenom-
enon to exercise its full effect on his mind, perceives,
that equilibrium is determined not by the weights
alone but also by their possible approach to and reces-
sion from the centre of the earth. Thus, while Q/2 de-
scends along the vertical height the distance h, Q as-
cends h along the inclined length, vertically, however^
only h/2 ; the result being that the products Q{h/2)
and {Q/2')h come out equal on both sides. The eluci-
dation that Galileo's observation affords and the light character
it diffuses, can hardly be emphasised strongly enough, observation
The observation is so natural and unforced, moreover,
that we admit it at once. What can appear simpler
. than that no motion takes place in a system of heavy

52 THE SCIENCE OF MECHANICS.

bodies when on the whole no heavy mass can descend.
Such a fact appears instinctively acceptable.
Comparison Galileo's conception of the inclined plane strikes

of it with . .

thatofste- US as much less ingenious than that of Stevinus, but
we recognise it as more natural and more profound. It
is in this fact that Galileo discloses such scientific great-
ness : that he had the intellectual audacity to see, in a
subject long before investigated, more than his prede-
cessors had seen, and to trust to his own perceptions.
With the frankness that was characteristic of him he
unreservedly places before the reader his own view,
together with the considerations that led him to it.

The Torn- 3. ToRRiCELLi, by the employment of the notion of

cellian ,

form of the "centre of gravity,*' has put Galileo's principle in a

principle. «j ^

form in which it appeals still more to our instincts, but
in which it is also incidentally applied by Galileo him-
self. According to Torricelli equilibrium exists in a
machine when, on a displacement being imparted to it,
the centre of gravity of the weights attached thereto
cannot descend. On the supposition of a displacement
in the inclined plane last dealt with, P^ let us say, de-
scends the distance A, in compensation wherefor Q
vertically ascends h . sin a. Assuming that the centre
of gravity does not descend, we shall have

P.h — ^.-^sina „ ^, xi/- a
^ j-y^ = 0, or Z'. ^ — ^ . // sin a = 0,

If the weights bear to one another some different pro-
portion, then the centre of gravity can descend when a
displacement is made, and equilibrium will not obtain.
We expect the state of equilibrium instinctively, when
- the centre of gravity of a system of heavy bodies can-

THE PRINCIPLES OF STA TICS. ^ 53

not descend. The Torricellian form of expression, how-
ever, contains in no respect more than the Galilean.

4. As with systems of pulleys and with the inclined The appu-
plane, so also the validity of *the principle of virtual the pnnci-
displacements is easily demonstrable for the other ma- other ma-
chines : for the lever, the wheel and axle, and the rest.

In a wheel and axle, for instance, with the radii R, r
and the respective weights P^ Q, equilibrium exists,
as we know, when PR = Qr. If we turn the wheel
and axle through the angle a, P will descend Pa, and
Q will ascend ra. According to the conception of
Stevinus and Galileo, when equilibrium exists, P. Pa
= Q . ra, which equation expresses the same thing as
the preceding one.

5. When we compare a system of heavy bodies in The cnte-
which motion is taking place, with a similar system state of
which is in equilibrium, the question forces itself upon *^"* ""™
us: What constitutes the difference of the two cases?

What is the factor operative here that determines mo-
tion, the factor that disturbs equilibrium, — the factor
that is present in the one case and absent in the other?
Having put this question to himself, Galileo discovers
that not only the weights, but also the distances of
their vertical descents (the amounts of their vertical
displacements) are the factors that determine motion.
Let us call P, P\ P" . . , the weights of a system of
heavy bodies, and A, K, h* , , , their respective, simul-
taneously possible vertical displacements, where dis-
placements downwards are reckoned as positive, and
displacements upwards as negative. Galileo finds
then, that the criterion or test of the state of equilib-
rium is contained in the fulfilment of the condition
Ph + />'// + P" >4" + . . . = 0. The sum Ph + P'h*
-j- P"h!*'\- ... is the factor that destroys equilibrium.

54 ^^^ SCIENCE OF MECHANICS,

the factor that determines motion. Owing to its im-
portance this sum has in recent times been character-
ised by the special designation work.
There is no 6. Whereas the earlier investigators, in the compari-
our choice son of cascs of equilibrium and cases of motion, directed
teria. their attention to the weights and their distances from

the axis of rotation and recognised the statical mo-
ments as the decisive factors involved, Galileo fixes
his attention on the weights and their distances of de-
scent and discerns work as the decisive factor involved.
It cannot of course be prescribed to the inquirer
what mark or criterion of the condition of equilibrium
he shall take account of, when several are present to
choose from. The result alone can determine whether
his choice is the right one. But if we cannot, for rea-
And all are SOUS already stated, regard the significance of the stat-
from the ical moments as given independently of experience, as

source. Something self-evident, no more can we entertain this
view with respect to the import of work. Pascal errs,
and many modern inquirers share this error with him,
when he says, on the occasion of applying the principle
of virtual displacements to fluids: '* Etant clair que c'est
la meme chose de faire faire un pouce de chemin k cent
livres d'eau, que de faire faire cent pouces de chemin
^ une livre d'eau. " This is correct only on the suppo-
sition that we have already come to recognise work as
the decisive factor ; and that it is so is a fact which
experience alone can disclose.
Illustration If we have an equal- armed, equally- weighted lever
ceding re- before US, we recognise the equilibrium of the lever as
the only effect that is uniquely determined, whether we
regard the weights and the distances or the weights
and the vertical displacements as the conditions that
determine motion. Experimental knowledge of this

THE PRINCIPLES OF STATICS. 55

or a similar character must, however, in the necessity of

the case precede any judgment of ours with regard to

the phenomenon in question. The particular way in

which the disturbance of equilibrium depends on the

conditions mentioned, that is to say, the significance

of the statical moment {PL) or of the work {Pk), is

even less capable of being philosophically excogitated

than the general fact of the dependence.

7. When two equal weights with equal and op- RsdncUon

posite possible displacements are opposed to eachenicuaot
'^ , '^ . , , . , .,., iheprinci-

other, we recognise at once the subsistence of equilib- pia to iba

ftlioplflr HOd

num. We might now be tempted to reduce the more>pBci.ic«io
general case of the weights P, P' with the capacities of
displacement^, A', where
Ph = P'h', to the sim-
pler case. Suppose we
have, for example, (Fig,
41) the weights 3 P and
^ Poaa. wheel and axle
with the radii 4 and 3.
We divide the weights
into equal portions of the

definite magnitude P, which we designate by a, b, c,
''■ 'i /' S- We then transport a, b, c to the level + 3,
and d, e, f to the level — 3. The weights will, of
themselves, neither enter on this displacement nor
will they resist it. We then take simultaneously the
weight g at the level 0 and the weight a at the level
-|- 3, push the first upwards to — i and the second
downwards to -f- 4, then again, and in the same way,
f to — 2 and ^ to 4- 4, f to — 3 and <: to + 4. To all
these displacements the weights offer no resistance,
nor do they produce them of themselves. Ultimately,
however, a, b, c (or 3/^ appear at the level -f- 4 and

56 THE SCIENCE OF MECHANICS,

The gen- //, ^, /, g (pr \P) at the level — 3. Consequently,
with respect also to the last-mentioned total displace-
ment, the weights neither produce it of themselves
nor do they resist it ; that is to say, given the ratio of
displacement here specified, and the weights will be
in equilibrium. The equation 4 . 3/* — 3 . ^F= 0 is,
therefore, characteristic of equilibrium in the case as-
sumed. The generalisation (Fh — P'h' = 0) is ob-
vious.
Thecondi- If we Carefully examine the reasoning of this case,
character we shall quite readily perceive that the inference in-
ence. "volved Cannot be drawn unless we take for granted
that the order of the operations performed and the path
by which the transferences are effected, are indifferent,
that is unless we have previously discerned that work
is determinative. We should commit, if we accepted
this inference, the same error that Archimedes com-
mitted in his deduction of the law of the lever ; as has
been set forth at length in a preceding section and
need not in the present case be so exhaustively dis-
cussed. Nevertheless, the reasoning we have pre-
sented is useful, in the respect that it brings palpably
home to the mind the relationship of the simple and
the complicated cases.
Theuniver- 8. The universal applicability of the principle of
bUiry^ofThe virtual displacements to all cases of equilibrium, was
Srstper-* perceived by John Bernoulli ; who communicated his
John Ber- discovery to Varignon in a letter written in 171 7. We
will now enunciate the principle in its most general
form. At the points A, Bf C , . . (Fig. 42) the forces
P, P\ P'* . . . are applied. Impart to the points any
infinitely small displacements 7^, v\ v'* . . . compatible
with the character of the connections of the points (so-
called virtual displacements), and construct the pro-

THE PRINCIPLES OF ST A TICS. 57

jections /, p\ p" of these displacements on the direc- General

. , , _,_ . . . , enunciation

tions of the forces. These projections we consider of the pnn-

positive when they fall in ji p >

the direction of the force, $^

and negative when they fall

in the opposite direction.

The products Fp, P* p\

/•"/", . . . are called virtual

moments, and in the two

cases just mentioned have K6-4a-

contrary signs. Now, the principle asserts, that for the

case of equilibrium Pp + /"/ + P" f + ...== 0, or

more briefly '^Pp = 0.

9. Let us now examine a few points more in detail. Detailed
Previous to Newton a force was almost universally tion of the
conceived simply as the pull or the pressure of a heavy
body. The mechanical researches of this period dealt
almost exclusively with heavy bodies. When, now,
in the Newtonian epoch, the generalisation of the idea
of force was effected, all mechanical principles known
to be applicable to heavy bodies could be transferred
at once to any forces whatsoever. It was possible to
replace every force by the pull of a heavy body on a
string. In this sense we may also apply the principle
of virtual displacements, at first discovered only for
heavy bodies, to any forces whatsoever.

Virtual displacements are displacements consistent Definition
with the character of the connections of a system and diBpiace-
with one another. If, for example, the two points of
a system, A and B^ at which forces act, are connected
(Fig. 43, i) by a rectangularly bent lever, free to re-
volve about C, then, if CB =:i2CAy all virtual dis-
placements of B and A are elements of the arcs of cir-
cles having C as centre ; the displacements of B are

58 T//E SCIENCE OF MECHANICS,

always double the displacements of Ay and both are in
every case at right angles to each other. If the points
Ay B (Fig. 43, 2) be connected by a thread of the length

/, adjusted to slip through

stationary rings at C and Z>,

then all those displacements

^ 2 ®-^ of A and B are virtual in

Fig. 43. which the points referred to

move upon or within two spherical surfaces described

with the radii r^ and r^ about C and D as centres,

where ''1 + ''2 + ^^ = ^•

The reason The use of infinitely small displacements instead of

of inaniteiyfinile displacements, such as Galileo assumed, is justi-

piacements. fied by the following consideration. If two weights

are in equilibrium on an inclined plane (Fig. 44), the

equilibrium will not be disturbed if the inclined plane,

at points at which it is not in immediate contact with

the bodies considered, passes into
a surface of a different form. The
essential condition is, therefore,
the momentary possibility of dis-
Fig. 44. placement in the momentary con-

figuration of the system. To judge of equilibrium we
must assume displacements vanishingly small and such
only ; as otherwise the system might be carried over
into an entirely different adjacent configuration, for
which perhaps equilibrium would not exist.
A Hmita- That the displacements themselves are not decisive

but only the extent to which they occur in the direc-
tions of the forces, that is only their projections on the
lines of the forces, was, in the case of the inclined plane,
perceived clearly enough by Galileo himself.

With respect to the expression of the principle, it
will be observed, that no problem whatever is .presented

tion

THE PRINCIPLES OF STATICS.

if all the material points of the system on which forces General re-
act, are independent of each other. Each point thus
conditioned can be in equilibrium only in the event
that it is not movable in the direction in which the force
acts. The virtual moment of each such point vanishes
separately. If some of the points be independent of
each other, while others in their displacements are de-
pendent on each other, the remark just made holds
good for the former ; and for the latter the fundamental
proposition discovered by Galileo holds, that the sum
of their virtual moments is equal to zero. Hence, the
sum-total of the virtual moments of all jointly is equal
to zero.

lo. Let us now endeavor to get some idea of the Examples,
significance of the principle, by the consideration of a
few simple examples that cannot be
dealt with by the ordinary method
of the lever, the inclined plane, and
the like.

The differential pulley of Wes-
ton (Fig. 45) consists of two coax-
ial rigidly connected cylinders of
slightly different radii r^ and r^
<r^. A cord or chain is passed
round the cylinders in the manner
indicated in the figure. If we pull
in the direction of the arrow with
the force P, and rotation takes place
through the angle ^, the weight Q attached below will
be raised. In the case of equilibrium there will exist
between the two virtual moments involved the equa-
tion

The differ-
ential pal-
ley of We»
ton.

Fig. 45.

Q^'-^<p = ^r^<p,oxF

''l— ''2

2r,

THE SCIENCE OF MECHANICS.

A suspend- A wheel and axle of weight Q (Fig. 46), which on
and aiie. the unroUing of a cord to which the weight P is at-
tached rolls itself up on a second cord
wound round the axle and rises, gives
for the virtual moments in the case of
equilibrium the equation

F{R^r)cp=Qrcp, or P= ^-^.

In the particular case R — r = 0, we
must also put, for equilibrium, Qr = 0, or,
for finite values of r, Q = 0. In reality the
string behaves in this case like a loop in
which the weight Q is placed. The lat-
ter can, if it be different from zero, continue to roll itself
downwards on the string without moving the weight P,
If, however, when R^^r, we also put ^ = 0, the re-
sult will he P=^, an indeterminate value. As a mat-
ter of fact, every weight P holds the apparatus in equi-
librium, since when R = r none can possibly descend.
A double A double cylinder (Fig. 47) of the radii r, R lies with

a^horizon" frictiou On a horizontal surface, and a force Q is brought

to bear on the string at-
tached to it. Calling the re-
sistance due to friction P,

_ equilibrium exists when

^^^^^^^^ />= {R-r/R) Q, If P>
FiK. 47. {R^r/R) Q, the cylinder,

on the application of the force, will roll itself up on
the string.

Roberval's Balance (Fig. 48) consists of a paral-
lelogram with variable angles, two opposite sides of
which, the upper and lower, are capable of rotation
about their middle points A, B. To the two remaining
sides, which are always vertical, horizontal rods are

Roberval's
balance.

THE PRINCIPLES OF ST A TICS.

fastened. If from these rods we suspend two equal

weights Py equilibrium will subsist independently of

the position of the points

of suspension, because on

displacement the descent

of the one weight is always

equal to the ascent of the

other.

At three fixed points A^
B, C(Fig. 49) let pulleys

be placed, over which three strings are passed loaded three knot
with equal weights and knotted dX O, In what posi- •*^"***
tion of the strings will equilibrium exist? We will call
the lengths of the three strings AO = s^, BO = s^,

Fig. 48-

Discuaaion
of the case
of equilib-
rium of

Fig. 49.

CO = s^, To obtain the equation of equilibrium, let
us displace the point O in the directions s^ and s^ the
infinitely small distances ds^ and 6s ^^ and note that by
so doing every direction of displacement in the plane
ABC (Fig. 50) can be produced. The sum of the vir-
tual moments is

Fig. 50.

2 — B6s^ cos a + Pds^ cos (^ + /^ I q

B6s
+ P6s^ — P6s^ cos/? + P6s^ cos {a -f /3)

[1 — cos or + cos (or + /^] 6s^ -f [1 — cos/3
+ cos (or + /?)] ^^3 = 0.

But since each of the displacements 6s ^, 6s ^ is ar-

THE SCIENCE OF MECHANICS.

bitrary, and each independent of the other, and may by
themselves be taken = 0, it follows that

Therefore

1 — cos a -f cos (or 4" /^ = ^
1 _ cos/? + cos («+/?) = 0.

cos a = cos fit

Remarks on

and each of the two equations may be replaced by

1 — cos a + cos 2a = 0;

or cos a = \y
wherefore a + /? = 120°

Accordingly, in the case of equilibrium, each of the
ingcase. Strings makes with the others angles of 120° ; which is,
moreover, directly obvious, since three equal forces can
be in equilibrium only when such an arrangement ex-
ists. This once known, we may find the position of
the point O with respect to ABC in a number of dif-
ferent ways. We may proceed for instance as follows.
We construct on AB, BC, CA, severally, as sides,
equilateral triangles. If we describe circles about these
triangles, their common point of intersection will be
the point O sought ; a result which easily follows from
the well-known relation of the angles at the centre and
circumference of circles.
The case of A bar OA (Fig. 51) is revolvable about O in the
voivabie plane of the paper and makes with a fixed straight line

OX the variable angle
a. At A there is ap-
plied a force B which

about one
of its ex-
tremities.

Fig. 51.

makes with OX the
angle y, and at B, on
a ring displaceable

along the length of the bar, a force Q, making with
OX the angle /3, We impart to the bar an infinitely

THE PRINCIPLES OF STATICS. 63

small rotation, in consequence of which B and A move The case of

a bar r©-

forward the distances 6s and 6s, at right angles to OA^ voivaWe

_ about one

and we also displace the ring the distance 6r along the of its ez-

, , tremities.

bar. The variable distance OB we will call r, and we
will let OA =: a. For the case of equilibrium we have
then

Q6r cos (/?—«) + Q6s sin (/?—«) +
Fds^ sin(a— ;^) = 0.

As the displacement 6r has no effect whatever on
the other displacements, the virtual moment therein
involved must, by itself, =0, and since dr may be of
any magnitude we please, the coefficient of this virtual
moment must also = 0. We have, therefore,

e cos (/? — a) = 0,

or when Q is different from zero,

/? — a=90°.

Further, in view of the fact that 6s ^^=1 {a/r) 6s, we
also have

rQ sin (J3 — a) -^ a P sin (a — y) = 0,

or since sin (J3 — a) = i,

rQ + aP sin {a — y) = 0;

wherewith the relation of the two forces is obtained.

II. An advantage, not to be overlooked, which Every ^en-
every general principle, and therefore also the prin- cipio^in°

VOlvCS All

ciple of virtual displacements, fur- 1 1 | ^ economy of

nishes, consists in the fact that it —^l * °"* •

saves us to a great extent the ne- c
cessity of considering every new par- ^ 1
ticular case presented. In the posses-
sion of this principle we need not, for Fig. 52.
example, trouble ourselves about the details of a ma-
chme. If a new machine say were so enclosed in a

64 THE SCIENCE OF MECHANICS.

box (Fig. 52), that only two levers projected as points
of application for the force P and the weight P\ and
we should find the simultaneous displacements of these
levers to be h and h\ we should know immediately that
in the case of equilibrium Fh = P' h\ whatever the
construction of the machine might be. Every principle
of this character possesses therefore a distinct econom-
ical value.
Farther ro- 12. We return to the general expression of the prin-
the general ciple of virtual displacements, in order to add a few

expression *■

of the prin- further remarks. If

at the pomts A, B,
C . . . , the forces

and Py p\ p'* ....

are the projections
^*8- 53. of infinitely small

mutually compatible displacements, we shall have for
the case of equilibrium

Pp + P'p' + P'p'' + . . . = 0.

If we replace the forces by strings which pass over
pulleys in the directions of the forces and attach thereto
the appropriate weights, this expression simply as-
serts that the centre of gravity of the system pf weights
as a whole cannot descend. If, however, in certain dis-
placements it were possible for the centre of gravity
to rise, the system would still be in equilibrium, as the
heavy bodies would not, of themselves, enter on any
Modifies- such motion. In this case the sum above given would
previous be negative, or less than zero. The general expression
condition, of the condition of equilibrium is, therefore,

When for every virtual displacement there exists

THE PRINCIPLES OF ST A TICS. 65

another rquai and opposite to it, as is the case for ex-
ample in the simple machines, we may restrict ourselves
to the upper sign, to the equation. For if it were pos-
sible for the centre of gravity to ascend in certain
displacements, it would also have to be possible, in
consequence of the assumed reversibility of all the vir-
tual displacements, for it to descend. • Consequently,
in the present case, a possible rise of the centre of
gravity is incompatible with equilibrium.

The question assumes a different aspect, however, The condi-

, , -. , ,, '% ^ rry tlOD 18, that

when the displacements are not all reversible. Two the sum of
bodies connected together by strings can approach moments
each other but cannot recede from each other beyond equal to or
the length of the strings. A body is able to slide or zero,
roll on the surface of another body ; it can move away
from the surface of the second body, but it cannot
penetrate it. In these cases, therefore, there are dis-
placements that cannot be reversed. Consequently,
for certain displacements a rise of the centre of gravity
may take place, while the contrary displacements, to
which the descent of the centre of gravity corresponds,
are impossible. We must therefore hold fast to the
more general condition of equilibrium, and say, the sum
of the virtual moments is equal to or less than zero.

13. Lagrange in his Analytical Mechanics attempted The La-

Kransiao

a deduction of the principle of virtual displacements, deduction

,., ... ., A'l ^ -n of theprin-

which we will now consider. At the points A, By cipic.
C . . . . (Fig. 54) the forces F, P\ P'* . . . . act. We
imagine rings placed at the points in question, and
other rings A\ B^^C* . . . . fastened to points lying in
the directions of the forces. We seek some common
measure Qji of the forces /*, P\ P** .... that enables
us to put :

THE SCIENCE OF MECHANICS,

Effected by
means of a
Bet of pul-
leys and a
single
weight

2«".f = /",

where «, «', «".... are whole numbers. Further, we
make fast to the ring A' a string, carry this string back
2lvA forth n times between A* and A^ then through B^^

«' times back and forth between B^ and B, then through
C\ «" times back and forth between C and C, and,
finally, let it drop at C\ attaching to it there the weight
QJT,, * As the string has, now, in all its parts the ten-
sion ^/2, we replace by these ideal pulleys all the
forces present in the system by the single force QJT,.
If then the virtual (possible) displacements in any given
configuration of the system are such that, these dis-
placements occurring, a descent of the weight Qfi. can
take place, the weight will actually descend and pro-
duce those displacements, and equilibrium therefore
will not obtain. But on the other hand, no motion
will ensue, if the displacements leave the weight Q/i
in its original position, or raise it. The expression of
this condition, reckoning the projections of the virtual
displacements in the directions of the forces positive,

THE PRINCIPLES OF STATICS, 67

and having regard for the number of the turns of the
string in each single pulley, is

2np + 2«>' + 2«"/' + : . . ':^ 0.

Equivalent to this condition, however, is the ex-
pression

2« 9.p + 2«' |-/ + 2«" -|/' + . . . < 0,
or

14. The deduction of Lagrange, if stripped of the The con-
rather odd fiction of the pulleys, really possesses con- tures oj La-
vincing features, due to the fact that the action of a deduction,
single weight is much more immediate to our expe-
rience and is more easily followed than the action of
several weights. Yet it is not proved by the Lagrangian
deduction that work is the factor determinative of the
disturbance of equilibrium, but is, by the employment
of the pulleys, rather assumed by it. As a matter of
fact every pulley involves the fact enunciated and rec-
ognised by the principle of virtual displacements. The
replacement of all the forces by a single weight that
does the same work, presupposes a knowledge of the
import of work, and can be proceeded with on this as-
sumption alone. The fact that some certain cases are it is not.
more familiar to us and more immediate to our expe- proof,
rience has as a necessary result that we accept them
without analysis and make them the foundation of our
deductions without clearly instructing ourselves as to
their real character.

It often happens in the course of the development
of science that a new principle perceived by some in-
quirer in connection with a fact, is not immediately
recognised and rendered familiar in its entire generality.

68 THE SCIENCE OF MECHANICS.

Theezpe- Then, every expedient calculated to promote these
ployed to ends, is, as is proper and natural, called into service.

support all . ,, - , . , . , , ... - , ,

newprin- All manner of facts, in which the principle, although
contained in them, has not yet been recognised by in-
quirers, but which from other points of view are more
familiar, are called in to furnish a support for the new
conception. It does not, however, beseem mature
science to allow itself to be deceived by procedures of
this sort. If, throughout all facts, we clearly see and dis-
cern a principle which, though not admitting of proof,
can yet be known to prevail^ we have advanced much
farther in the consistent conception of nature than if
we suffered ourselves to be overawed by a specious

Value of the demonstration. If we have reached this point of view,

Lagraogian

proof. we shall, it is true, regard the-Lagrangian deduction
with quite different eyes ; yet it will engage neverthe-
less our attention and interest, and excite our satis-
faction from the fact that it makes palpable the simi-
larity of the simple and complicated cases.

15. Maupertuis discovered an interesting proposi-
tion relating to equilibrium, which he communicated
to the Paris Academy in 1740 under the name of the
"Loi de repos." This principle was more fully dis-
cussed by EuLER in 1751 in the Proceedings of the
Berlin Academy. If we cause infinitely small displace-
Th^Loide ments in any system, we produce a sum of virtual mo-
'''^*' ments Fp + F'p' + P"/' +-..., which only reduces
to zero in the case of equilibrium. This sum is the
work corresponding to the displacements, or since for
infinitely small displacements it is itself infinitely small,
the corresponding element of work. If the displace-
ments are continuously increased till a finite displace-
ment is produced, the elements of the work will, by
summation, produce a finite amount of work. So, if we

THE PRINCIPLES OF STATICS, 69

Start from any given initial configuration of the system statement-

1 ^ ' r ^ r ,- . of the priu-

and pass to any given final configuration, a certain cipie.
amount of work will have to be done. Now Maupertuis
observed that the work done when a final configura-
tion is reached which is a configuration of equilibrium,
is generally a maximum or a minimum ; that is, if we
carry the system through the configuration of equilib-
rium the work done is previously and subsequently
less or previously and subsequently greater than at the
configuration of equilibrium itself. For the configura-
tion of equilibrium

pp + rp' + /"/• + . . . = 0,

that is, the element of the work or the differential (more
correctly the variation) of the work is equal to zero.
If the differential of a function can be put equal to
zero, the function has generally a maximum or mini-
mum value.

16. We can produce a very clear representation to Graphical
the eye of the import of Maupertuis's principle. oftheim-

We imagine the forces of a system replaced byprindpie.*
Lagrange's pulleys with the weight Q/2, We suppose
that each point of the system is restricted to movement
on a certain curve and that the motion is such that
when one point occupies a definite position on its curve
all the other points assume uniquely determined po-
sitions on their respective curves. The simple ma-
chines are as a rule systems of this kind. Now, while
imparting displacements to the system, we may carry
a vertical sheet of white paper horizontally over the
weight ^/2, while this is ascending and descending
on a vertical line, so that a pencil which it carries shall
describe a curve upon the paper (Fig. 55). When the
pencil stands at the points a, r, doi the curve, there are,

70 THE SCIENCE OF MECHANICS.

L- we see, adjacent positions in the system of points at
which the weight Ql'i will stand higher or lower than in
the configuration given. The weight will then, if the
system be left to itself, pass into this lower position and

displace the system with it Accordingly, under condi-
tions of this kind, equilibrium does not subsist. If
the pencil stands at e, then there exist only adjacent
configurations for which the weight Q|^ stands higher.
But of itself the system will not pass into the last-
named configurations. On the contrary, every dis-
placement in such a direction, will, by virtue of the
tendency of the weight to move downwards, be re-
versed. Stable equilibrium, Iherejore, is Ihc condition

■AKBtrtA- that corresponds to the lowest position of the weight or to
a maximum of work done in the system. If the pencil
stands at b, we see that every appreciable displace-
ment brings the weight QI2 lower, and that the weight
therefore will continue the displacement begun. But,
assuming infinitely small displacements, the pyencil
moves in the horizontal tangent at b, in which event
the weight cannot descend. Therefore, unstable equi-

labis tibrium is the stale that corresponds to the highest position
of the weight Qji, or to a minimum of work done in the
system. It will be -noted, however, that conversely

THE PRINCIPLES OF STATICS. 71

every case of equilibrium is not the correspondent of
a maximum or a minimum of work performed. If the
pencil is at/, at a point of horizontal contrary flexure,
the weight in the case of infinitely small displace-
ments neither rises nor falls. Equilibrium exists, al-
though the work done is neither a maximum nor a
minimum. The equilibrium of this case is the so-
called m/x^// equilibrium "^ : for some disturbances it is Mixed equi-
stable, for others unstable. Nothing prevents us from
regarding mixed equilibrium as belonging to the un-
stable class. When the pencil stands at g, where the
curve runs along horizontally a finite distance, equi-
librium likewise exists. Any small displacement, in
the configuration in question, is neither continued nor
reversed. This kind of equilibrium, to which likewise
neither a maximum nor a minimum corresponds, is
termed [neutral or] indifferent. If the curve described Nentrai
by Qji has a cusp pointing upwards, this indicates a*^""***''""
minimum of work done but no equilibrium (not even
unstable equilibrium). To a cusp pointing downwards
a maximum and stable equilibrium correspond. In the
last named case of equilibrium the sum of the virtual,
moments is not equal to zero, but is negative.

17. In the reasoning just presented, we have as-xhcpreced-
sumed that the motion of a point of a system on one tufn applied
curve determines the motion of all the other points oitoSlneSit'
the system on their respective curves. The movability *^"'*^"®"*
of the system becomes multiplex, however, when each
point is displaceable on a surface, in a manner such
that the position of one point on its surface determines

*This term is not used in English, because our writers bold tbat no
eqailibrinm is conceivable wbich is not stable or neutral for some possible
displacements. Hence what is called mixtd equilibrium in the text is called
onatable equilibrium by EuKlish writers, who deny the existence of equilibrium
mutable in every respect.— 7V0»#.

72 THE SCIENCE OF MECHANICS.

uniquely the position of all the other points on their
surfaces. In this case, we are not permitted to consider
the curve described by ^/2, but are obliged to picture
to ourselves a surface described by Q/2, If, to go a
step further, each point is movable throughout a space,
we can no longer represent to ourselves in a purely geo-
metrical manner the circumstances of the motion, by
means of the locus of Q/2. In a correspondingly higher
degree is this the case when the position of one of the
points of the system does not determine conjointly all
the other positions, but the character of the system's
motion is more multiplex still. In all these cases, how-
ever, the curve described by Q/2 (Fig. 55) can serve
us as a symbol of the phenomena to be considered. In
these cases also we rediscover the Maupertuisian pro-
positions.
Further ex- We have also supposed, in our considerations up to
the same this point, that constant forces, forces independent of
the position of the points of the system, are the forces
that act in the system. If we assume that the forces
do depend on the position of the points of the system
(but not on the time), we are no longer able to conduct

our operations with simple pulleys, but

must devise apparatus the force active in

which, still exerted by Q/2, varies with the

displacement : the ideas we have reached,

however, still obtain. The depth of the

descent of the weight Q/2 is in every case

the measure of the work performed, which

is always the same in the same configura-

F»K- 56. tion of the system and is independent of

the path of transference. A contrivance which would

develop by means of a constant weight a force varying

with the displacement, would be, for example, a wheel

THE PRINCIPLES OF STATICS. 73

and axle (Fig. 56) with a non-circular wheel. It would
not repay the trouble, however, to enter into the de-
tails of the reasoning indicated in this case, since we
perceive at a glance its feasibility.

18. If we know the relation that subsists between The prin-

ciple of
the work done and the so-called vis viva of a sys- courtivron.

tern, a relation established in dynamics, we arrive

easily at the principle communicated by Courtivron in

1 749 to the Paris Academy, which is this : For the

stable
configuration of ,, equilibrium, at which the

work done is a , the vis viva of the system,

mmimum •'

in motion, is also a . • in its transit through

' mmimum ^ ,

these configurations.

19. A heavy, homogeneous triaxial ellipsoid resting illustration
on a horizontal plane is admirably adapted to illustrate ous kinds of

... . equilibrium

the various classes of equilibrium. When the ellip-
soid rests on the extremity of its smallest axis, it is in
stable equilibrium, for any displacement it may suffer
elevates its centre of gravity. If it rest on its longest
axis, it is in unstable equilib-
rium. If the ellipsoid stand on
its mean axis, its equilibrium is
mixed. A homogeneous sphere
or a homogeneous right cylin-
der on a horizontal plane illus- Fi«- 57.
trates the case of indifferent equilibrium. In Fig. 57
we have represented the paths of the centre of gravity
of a cube rolling on a horizontal plane about one of its
edges. The position a of the centre of gravity is the
position of stable equilibrium, the position ^, the posi-
tion of unstable equilibrium.

74 THE SCIENCE OF MECHANICS,

The eaten- 20. We will now consider an example which at

81 ry.

first sight appears very complicated but is elucidated
at once by the principle of virtual displacements. John
and James Bernoulli, on the occasion of a conversa-
tion on mathematical topics during a walk in Basel,
lighted on the question of what form a chain would
take that was freely suspended and fastened at both
ends. They soon and easily agreed in the view that
the chain would assume that form of equilibrium at
which its centre of gravity lay in the lowest possible
position. As a matter of fact we really do perceive
that equilibrium subsists when all the links of the chain
have sunk as low as possible, when none can sink lower
without raising in consequence of the connections of
the system an equivalent mass equally high or higher.
When the centre of gravity has sunk as low as it pos-
sibly can sink, when all has happened that can happen,
stable equilibrium exists. The physical part of the
problem is disposed of by this consideration. The de-
termination of the curve that has the lowest centre of
gravity for a given length between the two points A^
Bt is simply a ma//ifma/ica/ problem. (See Fig. 58.)
Theprinci- 21. Collecting all that has been presented, we see,
plytherec- that there is contained in the principle of virtual dis-

ognition of , • % ^ • • • r r 1

a fact. placements simply the recognition of a fact that was
instinctively familiar to us long previousl)% only that
we had not apprehended it so precisely and clearly.
This fact consists in the circumstance that heavy
bodies, of themselves, move only downwards. If sev-
eral such bodies be joined together so that they can
suffer no displacement independently of each other,
they will then move only in the event that some heavy
mass is on the whole able to descend, or as the prin-
ciple, with a more perfect adaptation of our ideas to

THE PRINCIPLES OF STATICS,

76 THE SCIENCE OF MECHANICS,

What this the f acts, more exactly expresses it, only in the event
that work can be performed. If, extending the notion
of force, we transfer the principle to forces other than
those due to gravity, the recognition is again con-
tained therein of the fact that the natural occurrences
in question take place, of themselves, only in a definite
sense and not in the opposite sense. Just as heavy
bodies descend downwards, so differences of tempera-
ture and electrical potential cannot increase of their
own accord but only diminish, and so on. If occur-
rences of this kind be so connected that they can take
place only in the contrary sense, the principle then es-
tablishes, more precisely than our instinctive appre-
hension could do this, the factor work as determinative
and decisive of the direction of the occurrences. The
equilibrium equation of the principle may be reduced
in every case to the trivial statement, that when noth-
ing can happen nothing does happen,
Theprin- 22. It is important to' obtain clearly the perception,

Hgh?o? *that we have to deal, in the case of all principles,
view. merely with the ascertainment and establishment of a
fact. If we neglect this, we shall always be sensible
of some deficiency and will seek a verification of the
principle, that is not to be found. Jacobi states in his
Lectures on Dynamics that Gauss once remarked that
Lagrange's equations of motion had not been proved,
but only historically enunciated. And this view really
seems to us to be the correct one in regard to the prin-
ciple of virtual displacements.
The differ- The task of the early inquirers, who lay the foun-

ent tasks of,. . , .r- •!

early and of dations of any department of mvestigation, is entirely

inqufrers in different from that of those who follow. It is the busi-

mSn,*** ness of the former to seek out and to establish the

facts of most cardinal importance only; and, as history

THE PRINCIPLES OF STA TICS, 77

teaches, more brains are required for this than is gen-
erally supposed. When the most important facts are
once furnished, we are then placed in a position to
work them out deductively and logically by the meth-
ods of mathematical physics; we can then organise the
department of inquiry in question, and show that in the
acceptance of some one fact a whole series of others is
included which were not to be immediately discerned
in the first. The one task is as important as the other.
We should not however confound the one with the
other. We cannot prove by mathematics that nature
must be exactly what it is. But we can prove, that
one set of observed properties determines conjointly
another set which often are not directly manifest.

Let it be remarked in conclusion, that the princi- Every yen-
pie of virtual displacements, like every general prin- pie brings

witli it Qis-

ciple, brings with it, by the insight which it furnishes, iiiuaion-
disillusiontnent as well as elucidation. It brings with well as eia<
it disillusionment to the extent that we recognise in it
facts which were long before known and even instinct-
ively perceived, our present recognition being simply
more distinct and more definite ; and elucidation, in
that it enables us to see everywhere throughout the
most complicated relations the same simple facts.

RETROSPECT OF THE DEVELOPMENT OF STATICS.

I. Having passed successively in review the prin- Review of
ciples of statics, we are now in a position to take a whouf. ** *
brief supplementary survey of the development of the
principles of the science as a whole. This development,
falling as it does in the earliest period of mechanics,
— the period which begins in Grecian antiquity and

78 THE SCIENCE OF MECHANICS.

reaches its close at the time when Galileo and his
younger contemporaries were inaugurating modern me-
chanics,— illustrates in an excellent manner the pro-
cess of the formation of science generally. All con-
ceptions, all methods are here found in their simplest
form, and as it were in their infancy. These beginnings
The origin point unmistakably to their origin in the experiences of

of science.

the manual arts. To the necessity of putting these ex-
periences into communicable form and of disseminating
them beyond the confines of class and craft, science
owes its origin. The collector of experiences of this
kind, who seeks to preserve them in written form, finds
before him many different, or at least supposably differ-
ent, experiences. His position is one that enables him
to review these experiences more frequently, more vari-
ously, and more impartially than the individual work-
in gman, who is always limited to a narrow province.
The facts and their dependent rules are brought into
closer temporal and spatial proximity in his mind and
writings, and thus acquire the opportunity of revealing
The econo- their relationship, their connection, and their gradual
munication. transition the one into the other. The desire to sim-
plify and abridge the labor of communication supplies
a further impulse in the same direction. Thus, from
economical reasons, in such circumstances, great num-
bers of facts and the rules that spring from them are
condensed into a system and comprehended in a single
expression.
The gene- 2. A collcctor of this character has, moreover, op-

ral charttC'

ter of prin- portuuity to take note of some new aspect of the facts
before him — of some aspect which former observers
had not considered. A rule, reached by the observation
of facts, cannot possibly embrace the entire fact, in all
its infinite wealth, in all its inexhaustible manifoldness ;

THE PRINCIPLES OF STATICS. 79

on the contrary, it can furnish only a rough outline of
the fact, one-sidedly emphasising the feature that is of
importance for the given technical (or scientific) aim in
view. What aspects of a fact are taken notice of, will
consequently depend upon circumstances, or even on Their form

, . .in many as-

the caprice of the observer. Hence there is always op- pects, «cci-
portunity for the discovery of new aspects of the fact,
which will lead to the establishment of new rules of
equal validity with, or superior to, the old. So, for in-
stance, the weights and the lengths of the lever-arms
were regarded at first, by Archimedes, as the conditions
that determined equilibrium. Afterwards, by Da Vinci
and Ubaldi the weights and the perpendicular distances
from the axis of the lines of force were recognised as
the determinative conditions. Still later, by Galileo,
the weights and the amounts of their displacements,
and finally by Varignon the weights and the directions
of the pulls with respect to the axis were taken as the
elements of equilibrium, and the enunciation of the
rules modified accordingly.

3. Whoever makes a new observation of this kind, our Uabii-
and establishes such a new rule, knows, of course, our in the men-
liability to error in attempting mentally to represent atrucdon of
the fact, whether by concrete images or in abstract con-
ceptions, which we must do in order to have the mental
model we have constructed always at hand as a substi-
tute for the fact when the latter is partly or wholly in-
accessible. The circumstances, indeed, to which we
have to attend, are accompanied by so many other,
collateral circumstances, that it is frequently difficult
to single out and consider those that are essential to the
purpose in view. Just think how the facts of friction,
the rigidity of ropes and cords, and like conditions in
machines, obscure and obliterate the pure outlines of

8o THE SCIENCE OF MECHANICS.

Thisiiabii- the main facts. No wonder, therefore, that the discov-

ity impels

us to seek erer or verifier of a new rule, urged by mistrust of him-
of all new Self, seeks after a proof of the rule whose validity he

rules. . ,

believes he has discerned. The discoverer or verifier
does not at the outset fully trust in the rule ; or, it may
be, he is confident only of a part of it. So, Archimedes,
for example, doubted whether the effect of the action
of weights on a lever was proportional to the lengths of
the lever-arms, but he accepted without hesitation the
fact of their influence in some way. Daniel Bernoulli
does not question the influence of the direction of a
force generally, but only the form of its influence. As
a matter of fact, it is far easier to observe that a circum-
stance has influence in a given case, than to determine
what influence it has. In the latter -inquiry we are in
much greater degree liable to error. The attitude of the
investigators is therefore perfectly natural and defens-
ible.
The natural The proof of the correctness of a new rule can be
proof. attained by the repeated application of it, the frequent
comparison of it with experience, the putting of it to
the test under the most diverse circumstances. This
process would, in the natural course of events, get car-
ried out in time. The discoverer, however, hastens to
reach his goal more quickly. He compares the results
that flow from his rule with all the experiences with
which he is familiar, with all older rules, repeatedly
tested in times gone by, and watches to see if he do
not light on contradictions. In this procedure, the
greatest credit is, as it should be, conceded to the oldest
and most familiar experiences, the most thoroughly
tested rules. Our instinctive experiences, those gen-
eralisations that are made involuntarily, by the irresist-
ible force of tb^ iunumerable facts that press in upon

THE PRINCIPLES OF STATICS. 8i

US, enjoy a peculiar authority; and this is perfectly
warranted by the consideration that it is precisely the
elimination of subjective caprice and of individual er-
ror that is the object aimed at.

In this manner Archimedes proves his law of the illustration
lever, Stevinus his law of inclined pressure, Daniel ceding re-
Bernoulli the parallelogram of forces, Lagrange the
principle of virtual displacements. Galileo alone is
perfectly aware, with respect to the last- mentioned
principle, that his new observation and perception are
of equal rank with every former one — that it is derived
from the same source of experience. He attempts no
demonstration. Archimedes, in his proof of the prin-
ciple of the lever, uses facts concerning the centre of
gravity, which he had probably proved by means of the
very principle now in question ; yet we may suppose
that these facts were otherwise so familiar, as to be un-
questioned,— so familiar indeed, that it may be doubted
whether he remarked that he had employed them in
demonstrating the principle of the lever. The instinc-
tive elements embraced in the views of Archimedes and
Stevinus have been discussed at length in the proper
place.

4. It is quite in order, on the making of a new dis- The posi-

,, « • 1 tiontbatad-

covery, to resort to all proper means to brmg the new vanced sci-
rule to the test. When, however, after the lapse of a occupy,
reasonable period of time, it has been sufficiently often
subjected to direct testing, it becomes science to recog-
nise that any other proof than that has become quite
needless ; that there is no sense in considering a rule
as the better established for being founded on others
that have been reached by the very same method of
observation, only earlier ; that one well-considered and
tested observation is as good as another. To-day, we

82 THE SCIENCE OF MECHANICS,

should regard the principles of the lever, of statical
moments, of the inclined plane, of virtual displace-
ments, and of the parallelogram of forces as discovered
by equivalent observations. It is of no importance now^
that some of these discoveries were made directly, while
others were reached by roundabout ways and as de-
pendent upon other observations. It is more in keep-
ing, furthermore, with the economy of thought and with
insieht bet- the assthctics of science, directly to recognise a principle
dficiaidem- (say that of the statical moments) as the key to the un-

ODStration. ,,. r , 'i r r i ,

derstandmg of a// the facts of a department, and really
see how it pen^ades all those facts, rather than to hold
ourselves obliged first to make a clumsy and lame de-
duction of it from unobvious propositions that involve
the same principle but that happen to have become
earlier familiar to us. This process science and the in-
dividual (in historical study) may go through once for
all. But having done so both are free to adopt a more
convenient point of view.
The mis- 5- In fact, this mania for demonstration in science

mania for results in a rigor that is false and mistaken. Some pro-
^emons ra- p^gj^j^j^g ^^^ held to be possessed of more certainty

than others and even regarded as their necessary and
incontestable foundation ; whereas actually no higher,
or perhaps not even so high, a degree of certainty at-
taches to them. Even the rendering clear of the de-
gree of certainty which exact science aims at, is not at-
tained here. Examples of such mistaken rigor are to
be found in almost every text-book. The deductions
of Archimedes, not considering their historical value,
are infected with this erroneous rigor. But the most
conspicuous example of all is furnished by Daniel Ber-
noulli's deduction of the parallelogram of forces (^Com-
ment. Acad. Fetrop, T. I.).

THE PRINCIPLES OF STATICS, 83

6. As already seen, instinctive knowledge enjoys The char-

acter of in>
our exceptional confidence. No longer knowing how stinctWe

we have acquired it, we cannot criticise the logic by
which it was inferred. We have personally contributed
nothing to its production. It confronts us with a force
and irresistibleness foreign to the products of volun-
tary reflective experience. It appears to us as some-
thing free from subjectivity, and extraneous to us, al-
though we have it constantly at hand so that it is more
ours than are the individual facts of nature.

All this has often led men to attribute knowledge of its anthor-
this kind to an entirely different source, namely, to view latSy su-
it as existing a priori in us (previous to all experience). ^"™®'
That this opinion is untenable was fully explained in
our discussion of the achievements of Stevinus. Yet
even the authority of instinctive knowledge, however
important it may be for actual processes of develop-
ment, must ultimately give place to that of a clearly and
deliberately observed principle. Instinctive knowledge
is, after all, only experimental knowledge, and as such
is liable, we have seen, to prove itself utterly insuffi-
cient and powerless, when some new region of expe-
rience is suddenly opened up.

7. The true relation and connection of the different The tree re-
principles is the historical one. The one extends farther principles
in this domain, the other farther in that. Notwith- cai one.
standing that some one principle, say the principle of
virtual displacements, may control with facility a
greater number of cases than other principles, still

no assurance can be given that it will always maintain
its supremacy and will not be outstripped by some new
principle. All principles single out, more or less arbi-
trarily, now this aspect now that aspect of the same
facts, and contain an abstract summarised rule for the

t
84 THE SCIENCE OF MECHANICS,

refigurement of the facts in thought. We can never
assert that this process has been definitively completed.
Whosoever holds to this opinion, will not stand in the
way of the advancement of science.
Conception 8. Let US, in conclusion, direct our attention for a

of force in

statics. moment to the conception of force in statics. Force is
any circumstance of which the consequence is motion.
Several circumstances of this kind, however, each single
one of which determines motion, may be so conjoined
that in the result there shall be no motion. Now stat-
ics investigates what this mode of conjunction, in gen-
eral terms, is. Statics does not further concern itself
about the particular character of the motion condi-
tioned by the forces. The circumstances determinative
of motion that are best known to us, are our own vo-
The origin litioual acts — our innervations. In the motions which
tion of we ourselves determine, as well as in those to which
we are forced by external circumstances, we are always
sensible of a pressure. Thence arises our habit of rep-
resenting all circumstances determinative of motion as
something akin to volitional acts — as pressures. The
attempts we make to set aside this conception, as sub-
jective, animistic, and unscientific, fail invariably. It
cannot profit us, surely, to do violence to our own nat-
ural-born thoughts and to doom ourselves, in that re-
gard, to voluntary mental penury. We shall subse-
quently have occasion to observe, that the conception
referred to also plays a part in the foundation of dy-
namics.

We are able, in a great many cases, to replace the
circumstances determinative of motion, '^ich occur in
nature, by our innervations, and thus to reach the idea
of a gradation of the intensity of forces. But in the esti-
mation of this intensity we are thrown entirely on the

pressure.

THE PRINCIPLES OF STATICS. 85

resources of our memory, and are also unable to com- The com-
municate our sensations. Since it is possible, how-acterofaii
ever, to represent every condition that determines
motion by a weight, we arrive at the perception that
all circumstances determinative of motion (all forces)
are alike in character and may be replaced and meas-
ured by quantities that stand for weight. The meas-
urable weight serves us, as a certain, convenient, and
communicable index, in mechanical researches, just as
the thermometer in thermal researches is an exacter
substitute for our perceptions of heat. As has pre- The idea of
viously been remarked, statics cannot wholly rid itself auxiliary
of all knowledge of phenomena of motion. This par- statics,
ticularly appears in the determination of the direction
of a force by the direction of the motion which it would
produce if it acted alone. By the point of application
of a force we mean that point of a body whose motion
is still determined by the force when the point is freed
from its connections with the other parts of the body.

Force accordingly is any circumstance that de-Thegene-
termines motion : and its attributes may be stated asbutesof
follows. The direction of the force is the direction of
motion which is determined by that force, alone. The
point of application is that point whose motion is de-
termined independently of its coi|nections with the
system. The magnitude of the f Ace is that weight
which, acting (say, on a string) in |he direction deter-
mined, and applied at the point in question, determines
the same motion or maintains th^same equilibrium.
The other circumstances that modify the determination
of a motion, but by themselves alon^ are unable to pro-
duce it, such as virtual displacements, the arms of
levers, and so forth, may be termed collateral condi-
tions determinative of motion and equilibrium.

86 THE SCIENCE OF MECHANICS,

VI.

THE PRINCIPLES OF STATICS IN THEIR APPLICATION TO

FLUIDS.

No essen- I* The Consideration of fluids has not supplied stat-

ppinti°or ics with many essentially new points of view, yet nu-
volveifjn merous applications and confirmations of the principles
««u J^*^*- already known have resulted therefrom, and physical
experience has been greatly enriched by the investiga-
tions of this domain. We shall devote, therefore, a few
pages to this subject.

2. To Archimedes also belongs the honor of found-
ing the domain of the statics of liquids. To him we
owe the well-known proposition concerning the buoy-
ancy, or loss of weight, of bodies immersed in liquids,
of the discovery of which Vitruvius, De Architectural
Lib. IX, gives the following account :
vitruvius's "Though Archimcdes discovered many curious
Archime? ' ' matters that evince great intelligence, that which I am
covery.*" **about to mention is the most extraordinary. Hiero,
**when he obtained the regal power in Syracuse, hav-
**ing, on the fortunate turn of his affairs, decreed a
"votive crown of gold to be placed in a certain temple
**to the immortal gods, commanded it to be made of
** great value, and assigned for this purpose an appro-
*' priate weight of the metal to the manufacturer. The
** latter, in due time, presented the work to the king,
** beautifully wrought ; and the weight appeared to cor-
** respond with that of the gold which had been as-
** signed for it.

**But a report having been circulated, that some of
** the gold had been abstracted, and that the deficiency

THE PRINCIPLES OF STATICS, 87

'* thus caused had been supplied by silver, Hiero was The ac-

, , . 1 r 1 1 • i • 1 1 count of Vi-

<< indignant at the fraud, and, unacquainted with the trnyius.
"method by which the theft might be detected, re-
'* quested Archimedes would undertake to give it his
"attention. Charged with this commission, he by
"chance went to a bath, and on jumping into the tub,
"perceived that, just in the proportion that his body
"became immersed, in the same proportion the water
"ran out of the vessel. Whence, catching at the
" method to be adopted for the solution of the proposi-
"tion, he immediately followed it up, leapt out of the
" vessel in joy, and returning home naked, cried out
"with a loud voice that he had found that of which he
"was in search, for he continued exclaiming, in Greek,
^^evpijxa^ evprjHa, (I have found it, I have found it !)"

3. The observation which led Archimedes to his statement

. of the Ar-

proposition, was accordingly this, that a body im-chimedean
mersed in water must raise an equivalent quantity of
water ; exactly as if the body lay on one pan of a balance
and the water on the other. This conception, which
at the present day is still the most natural and the
most direct, also appears in Archimedes's treatises On
Floating Bodies, which unfortunately have not been
completely preserved but have in part been resiored
by F. Commandinus.

The assumption from which Archimedes starts
reads thus :

"It is assumed as the essential property of a liquid The Archi-
that in all uniform and continuous positions of its parts sumption,
the portion that suffers the lesser pressure is forced
upwards by that which suffers the greater pressure.
But each part of the liquid suffers pressure from the
portion perpendicularly above it if the latter be sinking
or suffer pressure from another portion."

THE SCIENCE OF MECHANICS,

Analysis of
the princi-
ple.

Fig. 59.

Archimedes now, to present the matter briefly,
conceives the entire spherical earth as fluid in consti-
tution, and cuts out of it pyramids the vertices of
which lie at the centre (Fig. 59). All these pyramids

must, in the case of equilib-
rium, have the same weight,
and the similarly situated
parts of the same must all
suffer the same pressure.
If we plunge a body a of
the same specific gravity as
water into one of the pyra-
mids, the body will com-
pletely submerge, and, in
the case of equilibrium, will supply by its weight the
pressure of the displaced water. The body ^, of less
specific gravity, can sink, without disturbance of equi-
librium, only to the point at which the water beneath
it suffers the same pressure from the weight of the
body as it would if the body were taken out and the
submerged portion replaced by water. The body r,
of a greater specific gravity, sinks as deep as it possibly
can. That its weight is lessened in the water by an
amount equal to the weight of the water displaced,
will be manifest if we imagine the body joined to
another of less specific gravity so that a third body is
formed having the same specific gravity as water,
which just completely submerges.
The state of 4. When in the sixteenth century the study of the
in the SIX- works of Archimedcs was again taken up, scarcely the

teenth cen- ... r i_ • ■!_ j ^ j rni.

tury. pnnciples of his researches were understood. The

complete comprehension of his deductions was at that
time impossible.

Stevinus rediscovered by a method of his own the

THE PRINCIPLES OF STATICS,

most important principles of hydrostatics and the de- The discov-

x- r- J eriesofSte-

ductions therefrom. It was principally two ideas from vinus.
which Stevinus derived his fruitful conclusions. The
one is quite similar to that relating to the endless
chain. The other consists in the assumption that the
solidification of a fluid in equilibrium does not disturb
its equilibrium.

Stevinus first lays down this principle. Any given The first
mass of water A (Fig. 60), immersed in water, is m tai pnnd-
equilibrium in all its parts. If A

Fig. 6a

were not supported by the sur-
rounding water but should, let us
say, descend, then the portion of
water taking the place of A and
placed thus in the same circum-
stances, would, on the same as-
sumption, also have to descend.
This assumption leads, therefore, to the establishment
of a perpetual motion, which is contrary to our ex-
perience and to our instinctive knowledge of things.

Water immersed in water loses accordingly its The second
whole weight. If, now, we imagine the surface of the tai princi-

ole

submerged water solidified, the vessel formed by this
surface, the vas superficiarium as Stevinus calls it, will
still be subjected to the same circumstances of pres-
sure. If empty^ the vessel so formed will suffer an
upward pressure in the liquid equal to the weight of the
water displaced. If we fill the solidified surface with
some other substance of any specific gravity we may
choose, it will be plain that the diminution of the
weight of the body will be equal to the weight of the
fluid displaced on immersion.

In a rectangular, vertically placed parallelepipedal
vessel filled with a liquid, the pressure on the horizontal

THE SCIENCE OF MECHANICS.

Stevinna's
deductioDft.

Galileo, in
the treat-
ment of this
subject, em-
ploys the
principle of
virtual dis-
placements

base is equal to the weight of the liquid. The pressure
is equal, also, for all parts of the bottom of the same
area. When now Stevinus imagines portions of the
liquid to be cut out and replaced by rigid immersed
bodies of the same specific gravity, or, what is the
same thing, imagines parts of the liquid to become so-
lidified, the relations of pressure in the vessel will not
be altered by the procedure. But we easily obtain in
this way a clear view of the law that the pressure on
the base of a vessel is independent of its form, as well
as of the laws of pressure in communicating vessels,
and so forth.

5. Galileo treats the equilibrium of liquids in com-
municating vessels and the problems connected there-
with by the help of the principle of virtual displace-
ments. NN (Fig. 61) being the
common level of a liquid in equilib-
rium in two communicating vessels,
Galileo explains the equilibrium
here presented by observing that in
the case of any disturbance the dis-
placements of the columns are to
each other in the inverse proportion
of the areas of the transverse sec-
tions and of the weights of the columns — that is, as
with machines in equilibrium. But this is not quite cor-
rect. The case does not exactly correspond to the
cases of equilibrium investigated by Galileo in ma-
chines, which present indifferent equilibrium. With
liquids in communicating tubes every disturbance of the
common level of the liquids produces an elevation of
the centre of gravity. In the case represented in Fig.
61, the centre of gravity S of the liquid displaced from
the shaded space in A is elevated to S\ and we may

Pig. 6x.

THE PRINCIPLES OF STATICS.

Fig. 6a.

regard the rest of the liquid as not having been moved.
Accordingly, in the case of equilibrium, the centre of
gravity of the liquid lies at its lowest possible point

6. Pascal likewise employs the principle of virtual The same

principle

displacements, but in a more correct manner, leaving made use of

, , by Pascal.

the weight of the liquid out of account and considering
only the pressure at the surface. If we imagine two
communicating vessels to be closed by pistons (Fig.
62), and these pistons loaded with
weights proportional to their surface-
areas, equilibrium will obtain, because
in consequence of the invariability of
the volume of the liquid the displace-
ments in every disturbance are in-
versely proportional to the weights.
For Pascal, accordingly, it follows, as a necessary con-
sequence, from the principle of virtual displacements,
that in the case of equilibrium every pressure on a su-
perficial portion of a liquid is propagated with undi-
minished effect to every other superficial portion, how-
ever and in whatever position it be placed. No objec-
tion is to be made to discovering the principle in this
way. Yet we shall see later on that the more natural
and satisfactory conception is to regard the principle as
immediately given.

7. We shall now, after this historical sketch, again Detailed
examine the most important cases of liquid equilibrium, don of the
and from such different points of view as may be con- " ^
venient.

The fundamental property of liquids given us by
experience consists in the flexure of their parts on the
slightest application of pressure. Let us picture to our-
selves an element of volume of a liquid, the gravity of
which we disregard — say a tiny cube. If the slightest

92 THE SCIENCE OF MECHANICS.

The funda- excess of pressure be exerted on one of the surfaces of

mental . .

proi>ert7of this cube, (which we now conceive, for the moment,
mobility of as a fixed geometrical locus, containing the fluid but

their parts.

not of its substance) the liquid (supposed to have pre-
viously been in equilibrium and at rest) will yield and
pass out in all directions through the other five surfaces
of the cube. A solid cube can stand a pressure on its
upper and lower surfaces different in magnitude from
that on its lateral surfaces ; or vice versa, A fluid cube,
on the other hand, can retain its shape only if the same
perpendicular pressure be exerted on all its sides. A
similar train of reasoning is applicable to all polyhe-
drons. In this conception, as thus geometrically eluci-
dated, is contained nothing but the crude experience
that the particles of a liquid yield to the slightest pres-
sure, and that they retain this property also in the in-
terior of the liquid when under a high pressure ; it
being observable, for example, that under the condi-
tions cited minute heavy bodies sink in fluids, and so on.
A second With the mobility of their parts liquids combine

the com- still another property, which we will now consider. Li-
of their vol- quids suffer through pressure a diminution of volume
which is proportional to the pressure exerted on unit
of surface. Every alteration of pressure carries along
with it a proportional alteration of volume and density.
If the pressure diminish, the volume becomes greater,
the density less. The volume of a liquid continues to
diminish therefore on the pressure being increased, till
the point is reached at which the elasticity generated
within it equilibrates the increase of the pressure.

8. The earlier inquirers, as for instance those of the
Florentine Academy, were of the opinion that liquids
were incompressible. In 1761, however, John Canton
performed an experiment by which the compressibility

THE PRINCIPLES OF STATICS. 93

of water was demonstrated. A thermometer glass is Ths £ni
filled wit4i water, boiled, and then sealed. (F\g. 63.)iioDo(ihe
The liquid reaches to a. But since the space above a is biiitv ot
airless, the liquid supports no atmospheric pres-
sure. If the sealed end be broken off, the liquid 1
will sink to i. Only a portion, however, of this L
displacement is to be placed to the credit of the '- W
compression of the liquid by atmospheric pres-
sure. For if we place the glass before breaking ^^
off the top under an air-pump and exhaust the ^^
chamber, the liquid will sink to c. This last phe- p.
nomenon is due to the fact that the pressure that
bears down on the exterior of the glass and diminishes
its capacity, is now removed. On breaking off the top,
this exterior pressure of the atmosphere is compensated
for by the interior pressure then introduced, and an
enlargement of the capacity of the glass again sets in.
The portion cb, therefore, answers to the actual com-
pression of the liquid by the pressure of the atmos-

The first, to institute exact experiments on the com- Tho eipeii
pressibility of water, was Oersted, who employed tooBr«t«ion
this end a very ingenious method. A
thermometer glass A (Fig. 64) is filled
with boiled water and is inverted, with
open mouth, into a vessel of mercury.
Near it stands a manometer tube B filled
with air and likewise inverted with open
mouth in the mercury. The whole ap-
paratus is then placed in a vessel filled i
with water, which is compressed by the 1
aid of a pump. By this means the water ^*- *»-
in A is also compressed, and the filament of quicksilver
which rises in the capillary tube of the thermometer-

94 THE SCIENCE OF MECHANICS.

glass indicates this compression. The alteration of
capacity which the glass A suffers in the present in-
stance, is merely that arising from the pressing to-
gether of its walls by forces which are equal on all sides.
The expert- The most delicate experiments on this subject have

ments of

Grassi. been conducted by Grassi with an apparatus con-
structed by Regnault, and computed with the assist-
ance of Lamp's correction-formulae. To give a tan-
gible idea of the compressibility of water, we will remark
that Grassi observed for boiled water at 0° under an
increase of one atmospheric pressure a diminution of
the original volume amounting to 5 in 100,000 parts.
If we imagine, accordingly, the vessel A to have the
capacity of one litre (1000 ccm.), and affix to it a cap-
illary tube of I sq. mm. cross-section, the quicksilver
filament will ascend in it 5 cm. under a pressure of
one atmosphere.
Surface- 9. Surface-pressure, accordingly, induces a physical

duces in alteration in a liquid (an alteration in density), which
alteration cau be detected by sufficiently delicate means — even
optical. We are always at liberty to think that por-
tions of a liquid under a higher pressure are more dense,
though it may be very slightly so, than parts under a
less pressure.
The impii- Let US imagine now, we have in a liquid (in the in-
ihii fact, terior of which no forces act and the gravity of which
we accordingly neglect) two portions subjected to un-
equal pressures and contiguous to one another. The
portion under the greater pressure, being denser, will
expand, and press against the portion under the less
pressure, until the forces of elasticity as le3sened on the
one side and increased on the other establish equilib-
rium at the bounding surface and both portions are
equally compressed.

THE PRINCIPLES OF STATICS, 95

If we endeavor, now, quantitatively to elucidate ourThestate-
mental conception of these two facts, the easy mobility these impU-
and the compressibility of the parts of a liquid, so that
they will fit the most diverse classes of experience,
we shall arrive at the following proposition : When
equilibrium subsists in a liquid, in the interior of which
no forces act and the gravity of which we neglect, the
same equal pressure is exerted on each and every equal
surface-element of that liquid however and wherever
situated. The pressure, therefore, is the same at all
points and is independent of direction.

Special experiments in demonstration of this prin-
ciple have, perhaps, never been instituted with the re-
quisite degree of exactitude. But the proposition has
by our experience of liquids been made very familiar,
and readily explains it.

10. If a liquid be enclosed in a vessel (Fig. 65) Preiimi-
which is supplied with a piston^, the cross-section marks to
of which is unit in area, and with a piston B which ion of Pas-

,..,.. , cal'sdedno-

for the time bemg is made station- ■z±-b>/ '*°"*

ary, and on the piston A a load p ^^E^3^^
be placed, then the same pressure ^^^^E^^^
/>, gravity neglected, will prevail ^^_-^i^^^
throughout all the parts of the vessel. _J^~- - - - ~ "^
The piston will penetrate inward and y^^^^^gr
the walls of the vessel will continue ^Ni^ '
to be deformed till the point is reached p»8- ^'

at which the elastic forces of the rigid and fluid bodies
perfectly equilibrate one another. If then we imagine
the piston B, which has the cross-section/, to be mov-
able, a force /. / alone will keep it in equilibrium.

Concerning Pascal's deduction of the proposition
before discussed from the principle of virtual displace-
ments, it is to be remarked that the conditions of dis-

96 THE SCIENCE OF MECHANICS,

Criticism of placement which he perceived hinge wholly upon the
daction. fact of the ready mobility of the parts and on the
equality of the pressure throughout every portion of
the liquid. If it were possible for a greater compression
to take place in one part of a liquid than in another,
the ratio of the displacements would be disturbed and
Pascal's deduction would no longer be admissible.
That the property of the equality of the pressure is a
property given in experience, is a fact that cannot be
escaped ; as we shall readily admit if we recall to mind
that the same law that Pascal deduced for liquids also
holds good for gases, where even approximately there
can be no question of a constant volume. This latter
fact does not afford any difficulty to our view ; but to
that of Pascal it does. In the case of the lever also, be
it incidentally remarked, the ratios of the virtual dis-
placements are assured by the elastic forces of the
lever-body, which do not permit of any great devia-
. tion from these relations.
Thebehav- u. We shall now consider the action of liquids un-

lourofli- , *

cniids under der the influence of gravity. The upper surface of a

of gravity. liquid in equilibrium is horizontal,

A7V(Fig. 66). This fact is at once
rendered intelligible when we re-
flect that every alteration of the sur-
face in question elevates the centre
of gravity of the liquid, and pushes
Fig. 66. ^i^g liquid mass resting in the shaded

space beneath NN and having the centre of gravity »S
into the shaded space above NN having the centre of
gravity S\ Which alteration, of course, is at once re-
versed by gravity.

Let there be in equilibrium in a vessel a heavy
liquid with a horizontal upper surface. We consider

THE PRINCIPLES OF STATICS, 97

(Fig. 67) a small rectangular parallelepipedon in the The con-
interior. The area of its horizontal base, we will say, is equUibriam
a, and the length of its vertical edges dh. The weight sabjected
of this parallelepipedon is therefore adhSy where s isuonofgrav-
its specific gravity. If the paral- ' ^'

p^dp

lelepipedon do not sink, this is
possible only on the condition that
a greater pressure is exerted on the
lower surface by the fluid than on
the upper. The pressures on the .
upper and lower surfaces we will ^**- ^•

respectively designate as ap and or (/ + dp). Equi-
librium obtains when adh,s = adp or dp/dh=:Sy
where h in the downward direction is reckoned as posi-
tive. We see from this that for equal increments of h
vertically downwards the pressure / must, correspond-
ingly, also receive equal increments. So that / =
hs-\- q\ and if ^, the pressure at the upper surface,
which is usually the pressure of the atmosphere, be-
comes = 0, we have, more simply, pz= hs, that is, the
pressure is proportional to the depth beneath the sur-
face. If we imagine the liquid to be pouring into a ves-
sel, and this condition of affairs not yet attained, every
liquid particle will then sink until the compressed par-
ticle beneath balances by the elasticity developed in it
the weight of the particle above.

From the view we have here presented it will be fur- Different
ther apparent, that the increase of pressure in a liquid tionrezist
takes place solely in the direction in which gravity Sneo? the*
acts. Only at the lower surface, at the base, of the ImSy!
parallelepipedon, is an excess of elastic pressure on the
part of the liquid beneath required to balance the
weight of the parallelepipedon. Along the two sides of
the vertical containing surfaces of the parallelepipedon,

98 THE SCIENCE OF MECHANICS,

the liquid is in a state of equal compression, since no
force acts in the vertical containing surfaces that would
determine a greater compression on the one side than
on the other.
Level sur- If we picture to ourselves the totality of all the
points of the liquid at which the same pressure / acts,
we shall obtain a surface — a so-called level surface. If
we displace a particle in the direction of the action of
gravity, it undergoes a change of pressure. If we dis-
place it at right angles to the direction of the action of
gravity, no alteration of pressure takes place. In the
latter case it remains on the same level surface, and
the element of the level surface, accordingly, stands at
right angles to the direction of the force of gravity.

Imagining the earth to be fluid and spherical, the
level surfaces are concentric spheres, and the directions
of the forces of gravity (the radii) stand at right angles
to the elements of the spherical surfaces. Similar ob-
servations are admissible if the liquid particles be acted
on by other forces than gravity, magnetic forces, for
example.
Their fano- The level surfaces afford, in a certain sense, a dia-
thought, gram of the force-relations to which a fluid is subjected;
a view further elaborated by analytical hydrostatics.

12. The increase of the pressure with the depth be-
low the surface of a heavy liquid may be illustrated by
a series of experiments which we chiefly owe to Pas-
cal. These experiments also well illustrate the fact,
that the pressure is independent of the direction. In
Fig. 68, I, is an empty glass tube g ground off at the
bottom and closed by a metal disc //, to which a
string is attached, and the whole plunged into a vessel
of water. When immersed to a sufficient depth we
may let the string go, without the metal disc, which is

THE PRINCIPLES OF STA TICS.

supported by the pressure of the liquid, falling. In 2, Pascal's ez-
the metal disc is replaced by a tiny column of mer- on the
cury. If (3) we dip an open siphon tube filled with Fiqufdi.
quicksilver into the water, we
shall see the quicksilver, in conse-
quence of the pressure at a^ rise
into the longer arm. In 4, we see
a tube, at the lower extremity of
which a leather bag filled with
quicksilver is tied : continued
immersion forces the quicksilver
higher and higher into tube. In
5, a piece of wood h is driven by
the pressure of the water into the
small arm of an empty siphon ^
tube. A piece of wood H (6) im-
mersed in mercury adheres to the h ■''■'''•"i-iiiiiiiii o^

bottom of the vessel, and is
pressed firmly against it for as
long a time as the mercury is
kept from working its way be-
neath it.

13. Once we have made quite
clear to ourselves that the pres-
sure in the interior of a heavy
liquid increases proportionally to
the depth below the surface, the
law that the pressure at the base
of a vessel is independent of its
form will be readily perceived.
The pressure increases as we de-
scend at an equal rate, whether the vessel (Fig. 69)
has the form abed ox ebcf. In both cases the walls
of the vessel where they meet the liquid, go on deforming

The pres-
sure at the
base of a
vessel inde-
endent of

fotm.

lOO

THE SCIENCE OF MECHANICS.

Elucida-
tion of this
fact.

Fig. 69.

The princi'
pie ox vir-
tual dis-
placements
applied to
the consid-
eration of
problems of
this class.

till the point is reached at which they equilibrate by the
elasticity developed in them the pressure exerted by the
fluid, that is, take the place as regards pressure of the

fluid adjoining. This fact is
a direct justification of Ste-
vinus's fiction of the solidi-
fied fluid supplying the place
of the walls of the vessel.
The pressure on the base
always remains P ■=. Ahs,
where A denotes the area of the base, h the depth of
the horizontal plane base below the level, and s the
specific gravity of the liquid.

The fact that, the walls of the vessel being neg-
lected, the vessels i, 2, 3 of Fig. 70 of equal base-
area and equal pressure-height weigh differently in the

balance, of course
in no wise con-
' tradicts the laws
of pressure men-
tioned. If we take
Fig. 70. into account the

lateral pressure, we shall see that in the case of i we
have left an extra component downwards, and in the
case of 3 an extra component upwards, so that on the
whole the resultant superficial pressure is always equal
to the weight.

14. The principle of virtual displacements is ad-
mirably adapted to the acquisition of clearness and
comprehensiveness in cases of this character, and we
shall accordingly make use of it. To begin with, how-
ever, let the following be noted. If the weight q (Fig.
71) descend from position i to position 2, and a weight
of exactly the same size move at the same time from

a •

« •

THE PRINCIPLES OF STATICS.

lOI

2 to 3, the work performed in this operation is qh^ + nary "?»■
qh^ z=q {h^ + ^g), the same, that is, as if the weight™*'*^*-
q passed directly from i to 3 and the weight at 2 re-
mained in its original position. The observation is
easily generalised.

y////////y///Mh

\(ik

^ V/y/y////////Adh

Fig. 71.

Fig. 72.

Let us consider a heavy homogeneous rectangular
parallelepipedon, with vertical edges of the length A,
base Ay and the specific gravity s (Fig. 72). Let this
parallelepipedon (or, what is the same thing, its centre
of gravity) descend a distance dh. The work done is
then Ahs.dhy or, also, A dhs.h. In the first expres-
sion we conceive the whole weight Ahs displaced the
vertical distance dh ; in the second we conceive the
weight Adhs as having descended from the upper
shaded space to the lower shaded space the distance h^

and leave out of account 1 T

the rest of the body.
Both methods of concep-
tion are admissible and
equivalent.

15. With the aid of
this observation we shall ____^^__
obtain a clear insight into p»8 73.

the paradox of Pascal, which consists of the following.
The vessel g (Fig. 73), fixed to a separate support and
consisting of a narrow upper and a very broad lower
cylinder, is closed at the bottom by a movable piston,

Pascal's
paradox.

I02

THE SCIENCE OF MECHANICS,

which, by means of a string passing through the axis
of the cylinders, is independently suspended from the
extremity of one arm of a balance. If g be filled with
water, then, despite the smallness of the quantity of
water used, there will have to be placed on the other
scale-pan, to balance it, several considerable weights,
the sum of which will he: Ahs^ where A is the piston-
area, // the height of the liquid, and s its specific grav-
ity. But if the liquid be frozen and the mass loosened
from the walls of the vessel, a very small weight will be
sufficient to preserve equilibrium.
The ezpia- Let US look to the virtual displacements of the two
the paradox cases (Fig. 74). In the first case, supposing the pis*
ton to be lifted a distance dh, the virtual moment is

A(ihs,h ox Ahs.dh, It thus
comes to the same thing,
dk whether we consider the mass

that the motion of the piston
'^=^^k displaces to be lifted to the
upper surface of the fluid
through the entire pressure-
height, or consider the entire weight Ahs lifted the
distance of the piston-displacement dh. In the second
case, the mass that the piston displaces is not lifted to
the upper surface of the fluid, but suffers a displace-
ment which is much smaller— the displacement, namely,
of the piston. If A, a are the sectional areas respect-
ively of the greater and the less cylinder, and k and /
their respective heights, then the virtual moment of the
present case is Adhs. k + ddhs , I = {Ak + «^) -f- d/i;
which is equivalent to the lifting of a much smaller
weight (^Ak -\- a/) s, the distance dA,

16. The laws relating to the lateral pressure of
liquids are but slight modifications of the laws of basal

Fig. 74.

THE PRINCIPLES OF STATICS.

103

pressure. If we have, for example, a cubical vessel The Uwa of
of I decimetre on the side, which is a vessel of litre pressure,
capacity, the pressure on any one of the vertical lateral
walls ABCDy when the vessel is filled with water^ is
easily determinable. The deeper the migratory element
considered descends beneath the surface, the greater
the pressure will be to which it is subjected. We easily
perceive, thus, that the pressure on a lateral wall is rep-
resented by a wedge of water A BCD HI resting upon
the wall horizontally
placed, where ID is at
right angles to BD and
ID = HC=Aa The
lateral pressure accor- ^^
dingly is equal to half
a kilogramme.

To determine the
point of application of the resultant pressure, conceive
ABCD again horizontal with the water-wedge resting
upon it. We cut off AK = BL = lACy draw the
straight line KL and bisect itzi M\ Mis the point of
application sought, for through this point the vertical
line cutting the centre of gravity of the wedge passes.

A plane inclined figure forming the base of a vessel The pres-
filled with a liquid, is divided into the elements a, a\ plane in-
a" . . . with the depths h, h\ A" . . . below the level of ^ *°* " "
the liquid. The pressure on the base is

If we call the total base-area A, and the depth of its
centre of gravity below the surface H, then

F»g- 75.

a+ a'+ a" + . , . A

whence the pressure on the base is AHs.

= H,

104

7'HE SCIENCE OF MECHANICS.

The deduc-
tion of the
principle of
Archime-
des may be
effected in
various
ways.

One meth-
od.

Another
method in-
volving the
principle of
virtual dis-
placementa

17. The principle of Archimedes can be deduced in
various ways. After the manner of Stevinus, let us
conceive in the interior of the liquid a portion of it
solidified. This portion now, as before, will be sup-
ported by the circumnatant liquid. The resultant of
the forces of pressure acting on the surfaces is accor-
dingly applied at the centre of gravity of the liquid dis-
placed by the solidified body, and is equal and opposite
to its weight. If now we put in the place of the solid-
ified liquid another different body of the same form, but
of a different specific gravity, the forces of pressure at
the surfaces will remain the same. Accordingly, there
now act on the body two forces, the weight of the body,
applied at the centre of gravity of the body, and the up-
ward buoyancy, the resultant of the surface-pressures,
applied at the centre of gravity of the displaced liquid.
The two centres of gravity in question coincide only in
the case of homogeneous solid bodies.

If we immerse a rectangular parallelepipedon of al-
titude h and base a, with edges vertically placed, in a
liquid of specific gravity s, then the pressure on the
upper basal surface, when at a depth k below the level
of the liquid is aks, while the pressure on the lower
surface is a (k -\- h) s. As the lateral pressures destroy
each other, an excess of pressure ahs upwards re-
mains ; or, where v denotes the volume of the paral-
lelepipedon, an excess v . s.

We shall approach nearest the fundamental con-
ception from which Archimedes started, by recourse to
the principle of virtual displacements. Let a paral-
lelepipedon (Fig. 76) of the specific gravity <r, base a,
and height h sink the distance dh. The virtual mo-
ment of the transference from the upper into the lower
shaded space of the figure will be adh . (T//. But while

THE PRINCIPLES OF STATICS.

105

this is done, the liquid rises from the lower into the up-
per space, and its moment is adhsh. The total vir-
tual moment is therefore ah{p — s)dh = (^p — ^) dhy
where p denotes the weight of the body and q the weight
of the displaced liquid.

Fig. 76.

FlR. 77.

18. The question might occur to us, whether the
upward pressure of a body in a liquid is affected by the
immersion of the latter in another liquid. As a fact,
this very question has been proposed. Let therefore
(Fig. 77) a body K be submerged in a liquid A and the
liquid with the containing vessel in turn submerged in
another liquid B, If in the determination of the loss
of weight in A it were proper to take account of the
loss of weight of A in B, then K*s loss of weight would
necessarily vanish when the fluid B became identical
with A. Therefore, K immersed in A would suffer a
loss of weight and it would suffer none. Such a rule
would be nonsensical.

With the aid of the principle of virtual displace-
ments, we easily comprehend the more complicated
cases of this character. If a body be first gradually
immersed in By then partly in B and partly in A,
finally in A wholly ; then, in the second case, consider-
ing the virtual moments, both liquids are to be taken
into account in the proportion of the volume of the
body immersed in them. But as soon as the body is
wholly immersed in A, the level of A on further dis-

l8 the buoy-
ancy of a
body in a
liquid af-
fected by
the immer-
sion of that
liquid in a
second
liquid?

The eluci-
dation of
more com-
plicated
cases of this
class.

io6

THE SCIENCE OF MECHANICS.

The Archi-
medean
principle il-
lustrated by
an experi-
ment.

The coun-
ter-experi-
ment.

Remarks on
the experi-
ment.

placement no longer rises, and therefore B is no longer
of consequence.

19. Archimedes*s principle may be illustrated by a
pretty experiment. From the one extremity of a scale-
beam (Fig. 78) we hang a hollow cube /T, and beneath

it a solid cube M^ which exactly fits into
the first cube. We put weights into the
opposite pan, until the scales are in
equilibrium. If now -^ be submerged
in water by lifting a vessel which stands
beneath it, the equilibrium will be dis-
turbed ; but it will be immediately re-
stored if /T, the hollow cube, be filled
with water.

A counter-experiment is the follow-
ing. H is left suspended alone at the
one extremity of the balance, and into
the opposite pan is placed a vessel of
water, above which on an independent
support J/ hangs by a thin wire. The scales are brought
to equilibrium. If now M be lowered until it is im-
mersed in the water, the equilibrium of the scales will
be disturbed ; but on filling H with water, it will be
restored.

At first glance this experiment appears a little para-
doxical. We feel, however, instinctively, that M can-
not be immersed in the water without exerting a pres-
sure that affects the scales. When we reflect, that the
level of the water in the vessel rises, and that the solid
body M equilibrates the surface- pressure of the water
surrounding it, that is to say represents and takes the
place of an equal volume of water, it will be found
that the paradoxical character of the experiment van-
ishes.

Fig. 78.

THE PRINCIPLES OF STATICS. 107

20. The most important statical principles have The gene-
been reached in the investigation of solid bodies. This pies of stai-
course is accidentally the historical one, but it is by no have been
means the only possible and necessary one. The dif- the invest!-
ferent methods that Archimedes, Stevinus, Galileo, and fluid bodies
the rest, pursued, place this idea clearly enough before
the mind. As a matter of fact, general statical princi-
ples, might, with the assistance of some very simple
propositions from the statics of rigid bodies, have been
reached in the investigation of liquids. Stevinus cer-
tainly came very near such a discovery. We shall stop
a moment to discuss the question.

Let us imagine a liquid, the weight of which we neg- The dis-
lect. Let this liquid be enclosed in a vessel and sub- illustration
jected to a definite pressure. A portion of the liquid, statement,
let us suppose, solidifies. On the closed surface nor-
mal forces act proportional to the elements of the area,
and we see without difficulty that their resultant will
always be = 0.

If we mark off by a closed curve a. portion of the
closed surface, we obtain, on either side of it, a non-
closed surface. All surfaces which are bounded by the
same curve (of double curvature) and on which forces
act normally (in the same sense) pro-
portional to the elements of the area,
have lines coincident in position for
the resultants of these forces.

Let us suppose, now, that a fluid
cylinder, determined by any closed
plane curve as the perimeter of its Fig. 79.

base, solidifies. We may neglect the two basal sur-
faces, perpendicular to the axis. And instead of the
cylindrical surface the closed curve simply may be con-
sidered. From this method follow quite analogous

io8 THE SCIENCE OF MECHANICS.

The dis- propositions for normal forces proportional to the ele-

cussion and . .

illustration ments of a plane curve.

statement. If the closed curve pass into a triangle, the con-
sideration will shape itself thus. The resultant normal
forces applied at the middle points of the sides of the
triangle, we represent in direction, sense, and magni-
tude by straight lines (Fig. 80). The
lines mentioned intersect at a point —
the centre of the circle described about
the triangle. It will further be noted,
Fig. 80. ^Yi^^ by the simple parallel displace-

ment of the lines representing the forces a triangle is
constructive which is similar and congruent to the
original triangle.
Thededac- Thence foUows this proposition :

tion of the * ■• r i • i , •

triangle of Any three forces, which, actmg at a pomt, are pro-
this method portional and parallel in direction to the sides of a tri-
angle, and which on meeting by parallel displacement
form a congruent triangle, are in equilibrium. We see
at once that this proposition is simply a different form
of the principle of the parallelogram of forces.

If instead of a triangle we imagine a polygon, we
shall arrive at the familiar proposition of the polygon
of forces.

We conceive now in a heavy liquid of specific gravity
K a portion solidified. On the element a of the closed
encompassing surface there acts a normal force axz,
where sis the distance of the element from the level of
the liquid. We know from the outset the result.
Similar de- If normal forces which are determined by axzy
another im- where a denotes an element of area and z its perpen-
position. dicular distance from a given plane E, act on a closed
surface inwards, the resultant will be V, x, in which ex-
pression F represents the enclosed volume. The

THE PRINCIPLES OF STA TICS, 109

resultant acts at the centre of gravity of the volume,

is perpendicular to the plane mentioned, and is directed

towards this plane.

Under the same conditions let a rigid curved surface The propo-
sition here
be bounded by a plane curve, which encloses on the deduced, a

special case

plane the area A. The resultant of the forces acting of Greens

, Theorem.

on the curved surface is R, where

^2 = {AZxy + ( Vxy — AZVk^ cos V,

in which expression Z denotes the distance of the
centre of gravity of the surface A from E^ and v the
normal angle of E and A.

In the proposition of the last paragraph mathe-
matically practised readers will have recognised a par-
ticular case of Green's Theorem, which consists in the
reduction of surface-integrations to volume-integra-
tions or vice versa.

We may, accordingly, see into the force-system of a The impii-

n •* * *i'i. * 'r 1 J c '. cations of

fluid in equilibrium, or, if you please, see out of it, sys- the view
tems of forces of greater or less complexity, and thus *^"*
reach by a short path propositions a posteriori. It is a
mere accident that Stevinus did not light on these
propositions. The method here pursued corresponds
exactly to his. In this manner new discoveries can
still be made.

2 1 . The paradoxical results that were reached in Fruitful re-
the investigation of liquids, supplied a stimulus to fur- investiga-
ther reflection and research. It should also not be left domain,
unnoticed, that the conception of a physico-mechanical
continuum was first formed on the occasion of the in-
vestigation of liquids. A much freer and much more
fruitful mathematical mode of view was developed
thereby, than was possible through the study even of

no THE SCIENCE OF MECHANICS

systems of several solid bodies. The origin, in fact,
of important modern mechanical ideas, as for instance
that of the potential, is traceable to this source.

vn.

THE PRINCIPLES OF STATICS IN THEIR APPLICATION TO

GASEOUS BODIES.

Character I . The Same views that subserve the ends of science

of this de-
partment of in the investigation of liquids are applicable with but

slight modifications to the investigation of gaseous
bodies. To this extent, therefore, the investigation of
gases does not afford mechanics any very rich returns.
Nevertheless, the first steps that were taken in this
province possess considerable significance from the
point of view of the progress of civilisation and high
import for science generally.
Theeius- Although the ordinary man has abundant oppor*

iu subject- t unity, by his experience of the resistance of the air, by
"* "■ the action of the wind, and the confinement of air in
bladders, to perceive that air is of the nature of a body,
yet this fact manifests itself infrequently, and never in
the obvious and unmistakable way that it does in the
case of solid bodies and fluids. It is known, to be sure,
but is not sufficiently familiar to be prominent in popu-
lar thought. In ordinary life the presence of the air is
scarcely ever thought of.
The effect Although the ancients, as we may learn from the
disclosures accouuts of Vitruvius, possessed instruments which,
ince* ^^^^ like the so-called hydraulic organs, were based on the
condensation of air, although the invention of the air-
gun is traced back to Ctesibius, and this instrument
was also known to Guericke, the notions which people
held with regard to the nature of the air as late even

THE PRINCIPLES OF STA TICS.

112 THE SCIENCE OF MECHANICS.

as the seventeenth century were exceedingly curious
and loose. We must not be surprised, therefore, at the
intellectual commotion which the first more important
experiments in this direction evoked. The enthusiastic
description which Pascal gives of Boyle's air-pump ex-
periments is readily comprehended, if we transport our-
selves back into the epoch of these discoveries. What
indeed could be more wonderful than the sudden dis-
covery that a thing which we do not see, hardly feel,
and take scarcely any notice of, constantly envelopes
us on all sides, penetrates all things ; that it is the most
important condition of life, of combustion, and of gi-
gantic mechanical phenomena. It was on this occa-
sion, perhaps, first made manifest by a great and strik-
ing disclosure, that physical science is not restricted
to the investigation of palpable and grossly sensible
processes.
The views 2. In Galileo's time philosophers explained the

entertained -i / ^' ^i_ ^' r ■ j

on this sub- phenomenon of suction, the action of syringes and
Ico's time.* pumps by the so-called horror vacui — nature's abhor-
rence of a vacuum. Nature was thought to possess
the power of preventing the formation of a vacuum by
laying hold of the first adjacent thing, whatsoever it
was, and immediately filling up with it any empty space
that arose. Apart from the ungrounded speculative
element which this view contains, it must be conceded,
that to a certain extent it really represents the phe-
nomenon. The person competent to enunciate it must
actually have discerned some principle in the phenom-
enon. This principle, however, does not fit all cases.
Galileo is said to have been greatly surprised at hearing
of a newly constructed pump accidentally supplied
with a very long suction-pipe which was not able to
raise water to ^ height of more than eighteen Italiaa

THE PRINCIPLES OF STATICS, 113

elJs. His first thought was that the horror vacui (or the
resistenza del vacuo) possessed a measurable power. The
greatest height to which water could be raised by suc-
tion he called altezza limitatissima. He sought, more-
over, to determine directly the weight able to draw out
of a closed pump-barrel a tightly fitting piston resting
on the bottom.

3. ToRRiCELLi hit upon the idea of measuring the Tomceiii'a
resistance to a vacuum by a column of mercury mstead
of a column of water, and he expected to obtain a col-
umn of about ^ of the length of the water column.
His expectation was confirmed by the experiment per-
formed in 1643 by Viviani in the well-known manner,
and which bears to-day the name of the Torricellian
experiment. A glass tube somewhat over a metre in
length, sealed at one end and filled with mercury, is
stopped at the open end with the finger, inverted in a
dish of mercury, and placed in a vertical position. Re-
moving the finger, the column of mercury falls and re-
mains stationary at a height of about 76 cm. By this
experiment it was rendered quite probable, that some
very definite pressure forced the fiuids into the vacuum.
What pressure this was, Torricelli very soon divined.

Galileo had endeavored, some time before this, to Galileo's
determine the weight of the air, by first weighing a wefi^^air?
glass bottle containing nothing but air and then again
weighing the bottle after the air had been partly ex-
pelled by heat. It was known, accordingly, that the
air was heavy. But to the majority of men the horror
vacui and the weight of the air were very distantly
connected notions. It is possible that in Torricelli's
case the two ideas came into sufficient proximity to
lead him to the conviction that all phenomena ascribed
to the horror vacui were explicable in a simple and

114 THE SCIENCE OF MECHANICS,

Atmospher- logical manner by the pressure exerted by the weight

discovered of a fluid columu — a column of air. Torricelli discov-

ceiii. ered, therefore, the pressure of the atmosphere ; he also

first observed by means of his column of mercury the

variations of the pressure of the atmosphere.

4. The news of Torricelli's experiment was circu-
lated in France by Mersenne, and came to the knowl-
edge of Pascal in the year 1644. The accounts of the
theory of the experiment were presumably so imper-
fect that Pascal found it necessary to reflect indepen-
dently thereon. iJPesanieur de Pair. Paris, 1663.)
pascars ex- He repeated the experiment with mercury and with

perimenis.

a tube of water, or rather of red wine, 40 feet in length.
He soon convinced himself by inclining the tube that
the space above the column of fluid was really empty ;
and he found himself obliged to defend this view against
the violent attacks of his countrymen. Pascal pointed
out an easy way of producing the vacuum which they
regarded as impossible, by the use of a glass syringe,
the nozzle of which was closed with the finger under
water and the piston then drawn back without much
difficulty. Pascal showed, in addition, that a curved
siphon 40 feet high filled with water does not flow, but
can be made to do so by a sufficient inclination to the
perpendicular. The same experiment was made on a
smaller scale with mercury. The same siphon flows
or does not flow according as it is placed in an inclined
or a vertical position.

In a later performance, Pascal refers expressly to
the fact of the weight of the atmosphere and to the
pressure due to this weight. He shows, that minute
animals, like flies, are able, without injury to them-
selves, to stand a high pressure in fluids, provided only
the pressure is equal on all sides ; and he applies this

THE PRINCIPLES OF STATICS,

at once to the case of fishes and of animals that live in The anai-
the air. Pascal's chief merit, indeed, is to have estab- Uquid and
lished a complete analogy between the phenomena con- ic pressure,
ditioned by liquid pressure (water- pressure) and those
conditioned by atmospheric pressure.

5. By a series of experiments Pascal shows that
mercury in consequence of atmospheric pressure rises
into a space containing no air in the same way that,
in consequence of water-pressure, it rises into a space
containing no water. If into a deep ves-
sel filled with water (Fig. 81) a tube be c^
sunk at the lower end of which a bag of
mercury is tied, but so inserted that the
upper end of the tube projects out of the
water and thus contains only air; then
the deeper the tube is sunk into the water
the higher will the mercury, subjected Fig. 81.
to the constantly increasing pressure of the water, as-
cend into the tube. The experiment can also be made,
with a siphon-tube, or with a tube open at its lower end.

Undoubtedly it was the attentive consideration of The height

.of moaii^

this very phenomenon that led Pascal to the idea that tains deter-

mined by

the barometer-column must necessarily stand lower atthebarom-
the summit of a mountain than at its base, and that
it could accordingly be employed to determine the
height of mountains. He communicated this idea to
his brother-in-law, Perier, who forthwith successfully
performed the experiment on the summit of the Puy
de Ddme. (Sept. 19, 1648.)

Pascal referred the phenomena connected with ad- Adhesion

plates.

hesion-plates to the pressure of the atmosphere, and
gave as an illustration of the principle involved the re-
sistance experienced when a large hat lying flat on a
table is suddenly lifted. The cleaving of wood to the

ii6

THE SCIENCE OF MECHANICS.

A siphon
which acts
by water-
pressare.

Pascal's
modifica-
tion of the
Torricelli-
an experi-
ment.

Fig. 8a.

bottom of a vessel of quicksilver is a phenomenon of
the same kind.

Pascal imitated the flow produced in a siphon by
atmospheric pressure, by the use of water- pressure.

The two open unequal arms a and
^ of a three-armed tube ab c (Fig.
82) are dipped into the vessels of
mercury e and d. If the whole
arrangement then be immersed in
a deep vessel of water, yet so that
the long open branch shall always
project above the upper surface,
the mercury will gradually rise in
the branches a and 3, the columns
finally unite, and a Stream begin to flow from the vessel
d to the vessel e through the siphon-tube open above
to the air.

PI J The Torricellian experiment was modi-

fied by Pascal in a very ingenious manner.
A tube of the form abed (Fig. 83), of
double the length of an ordinary barom-
eter-tube, is filled with mercury. The
openings a and b are closed with the fin-
gers and the tube placed in a dish of
mercury with the end a downwards. If
now a be opened, the mercury \vi ed will
all fall into the expanded portion at r, and
the mercury xn ab will sink to the height
of the ordinary barometer-column. A vac-
uum is produced at b which presses the
finger closing the hole painfully inwards.
If b also be opened the column \n ab will
sink completely, while the mercury in the expanded
portion r, being now exposed to the pressure of the

Fig. 83.

THE PRINCIPLES OF STATICS. 117

atmosphere, will rise in cd Xo the height of the barom-
eter-column. Without an air-pump it was hardly pos-
sible to combine the experiment and the counter-
experiment in a simpler and more ingenious manner
than Pascal thus did.

6. With regard to Pascal's mountain-experiment, Snppie-

, mentary re-

we shall add the following brief supplementary remarks, marks on
Let bf. be the height of the barometer at the level of mountain-

experiment

the sea, and let it fall, say, at an elevation of m metres,
to kb^y where >& is a proper fraction. At a further eleva-
tion of m metres, we must expect to obtain the barom-
eter-height k ,kb^^ since we here pass through a stratum
of air the density of which bears to that of the first the
proportion oi k \\. If we pass upwards to the altitude
h = n , m metres, the barometer-height corresponding
thereto will be

h kn hr.T^ log /^— log ^

Of.-=: RT . o^ox n-=. . — or

\ogk

The principle of the method is, we see, a very simple
one ; its difficulty arises solely from the multifarious
collateral conditions and corrections that have to be
looked to.

7. The most original and fruitful achievements in The expert-
the domain of aerostatics we owe to Otto von GuE-SSovon
RiCKE. His experiments appear to have been suggested
in the main by philosophical speculations. He pro-
ceeded entirely in his own way ; for he first heard of
the Torricellian experiment from Valerianus Magnus
at the Imperial Diet of Ratisbon in 1654, where he dem-
onstrated the experimental discoveries made by him
about 1650. This statement is confirmed by his method

ii8 THE SCIENCE OF MECHANICS.

of constructing a water-barometer which was entirely
different from that of Torricelli.
Thehistori- Guericke's book {Experimenta nova, ut vocantur,
Guericke'8 Magdeburgica, Amsterdam. 1672) makes us realise
the narrow views* men took in his time. The fact that
he was able gradually to abandon these views and to
acquire broader ones by his individual endeavor speaks
favorably for his intellectual powers. We perceive
with astonishment how short a space of time separates
us from the era of scientific barbarism, and can no lon-
ger marvel that the barbarism of the social order still
so oppresses us.
Its specula- In the introduction to this book and in various other
ter. places, Guericke, in the midst of his experimental in-

vestigations, speaks of the various objections to the
Copernican system which had been drawn from the
Bible, (objections which he seeks to invalidate,) and
discusses such subjects as the locality of heaven, the
locality of hell, and the day of judgment. Philoso-
phemes on empty space occupy a considerable portion
of the work.
Guericke'8 Guericke regards the air as the exhalation or odor
the air. of bodies, which we do not perceive because we have
been accustomed to it from childhood. Air, to him,
is not an element. He knows that through the effects of
heat and cold it changes its volume, and that it is
compressible in Hero's Ball, or Pila Heronis\ on the
basis of his own experiments he gives its pressure at
20 ells of water, and expressly speaks of its weight, by
which flames are forced upwards.

8. To produce a vacuum, Guericke first employed
a wooden cask filled with water. The pump of a fire-
engine was fastened to its lower end. The water, it
was thought, in following the piston and the action of

THE PRINCIPLES OF STATICS.

ricks'i Flnl EipariuwliU. ifixptrim. UapUi.y

I20 THE SCIENCE OF MECHANICS,

His at- gravity, would fall and be pumped out. Guericke ex-

tcinpts to

produce a pectcd that empty space would remain. The fastenmgs

vacuum. . .

of the pump repeatedly proved to be too weak, since m
consequence of the atmospheric pressure that weighed
on the piston considerable force had to be applied to
move it. On strengthening the fastenings three power-
ful men finally accomplished the exhaustion. But,
meantime the air poured in through the joints of the
cask with a loud blast, and no vacuum was obtained.
In a subsequent experiment the small cask from which
the water was to be exhausted was immersed in a larger
one, likewise filled with water. But in this case, too, the
water gradually forced its way into the smaller cask.
His 6nai Wood having proved in this way to be an unsuit-

SQCCOSS V* * ^

able material for the purpose, and Guericke having re-
marked in the last experiment indications of success,
the philosopher now took a large hollow sphere of
copper and ventured to exhaust the air directly. At
the start the exhaustion was successfully and easily
conducted. But after a few strokes of the piston, the
pumping became so difficult that four stalwart men
{viri quadrati), putting forth their utmost efforts, could
hardly budge the piston. And when the exhaustion
had gone still further, the sphere suddenly collapsed,
with a violent report. Finally by the aid of a copper
vessel of perfect spherical form, the production of the
vacuum was successfully accomplished. Guericke de-
scribes the great force with which the air rushed in on
the opening of the cock.

9. After these experiments Guericke constructed
an independent air-pump. A great glass globular re-
ceiver was mounted and closed by a large detachable
tap in which was a stop-cock. Through this opening
the objects to be subjected to experiment were placed

THE PRINCIPLES OF STATICS. iii

in the receiver. To secure more perfect closure the c
receiver was made to stand, with its stop-cock under
water, on a tripod, beneath which the pump proper was

Cuaricte'i Air-pump. (Eiftrim. Magdii.)

placed. Subsequently, separate receivers, connected
with the exhausted sphere, were also employed in the
experiments.

122 THE SCIENCE OF MECHANICS,

ThecarioM The phenomena which Guericke observed with this
observed by apparatus are manifold and various. The noise which
the air- Water in a vacuum makes on striking the sides of the
glass receiver, the violent rush of air and water into
exhausted vessels suddenly opened, the escape on ex-
haustion of gases absorbed in liquids, the liberation of
their fragrance, as Guericke expresses it, were imme-
diately remarked. A lighted candle is extinguished
on exhaustion, because, as Guericke conjectures, it
derives its nourishment from the air. Combustion, as
his striking remark is, is not an annihilation, but a
transformation of the air.

A bell does not ring in a vacuum. Birds die in it.
Many fishes swell up, and finally burst. A grape is kept
fresh in vacuo for over half a year.

By connecting with an exhausted cylinder a long
tube dipped in water, a water- barometer is constructed.
The column raised is 19-20 ells high; and Von Guericke
explained all the effects that had been ascribed to the
horror vacui by the principle of atmospheric pressure.
An important experiment consisted in the weighing
of a receiver, first when filled with air and then when
exhausted. The weight of the air was found to vary
with the circumstances ; namely, with the temperature
and the height of the barometer. According to Gue-
ricke a definite ratio of weight between air and water
does not exist.
The experi- But the deepest impression on the contemporary

mentsrelat-

ing to at- world was made by the experiments relating to atmos-

mospheric

pressure, pheric pressure. An exhausted sphere formed of two
hemispheres tightly adjusted to one another was rent
asunder with a violent report only by the traction of
sixteen horses. The same sphere was suspended from

THE PRINCIPLES OF STATICS. 123

a beam, and a heavily laden scale-pan was attached to
the lower half.

The cylinder of a large pump is closed by a piston.
To the piston a rope is tied which leads over a pulley
and is divided into numerous branches on which a
great number of men pull. The moment the cylinder is
connected with an exhausted receiver, the men at the
ropes are thrown to the ground. In a similar manner
a huge weight is lifted.

Guericke mentions the compressed-air gun as some- Guericke's
thing already known, and constructs independently an
instrument that might appropriately be called a rari-
fied-air gun. A bullet is driven by the external atmos-
pheric pressure through a suddenly exhausted tube,
forces aside at the end of the tube a leather valve which
closes it, and then continues its flight with a consider-
able velocity.

Closed vessels carried to the summit of a mountain
and opened, blow out air ; carried down again in the
same manner, they suck in air. From these and other
experiments Guericke discovers that the air is elastic.

10. The investigations of Guericke were continued The invead-
by an Englishman, Robert Boyle.* The new experi- Robert
ments which Boyle had to supply were few. He ob-
serves the propagation of light in a vacuum and the
action of a magnet through it ; lights tinder by means
of a burning glass ; brings the barometer under the re-
ceiver of the air-pump, and was the first to construct
a balance-manometer ['*the statical manometer"].
The ebullition of heated fluids and the freezing of water
on exhaustion were first observed by him.

Of the air-pump experiments common at the present
day may also be mentioned that with falling bodies,

* And published by him in 1660, before the work of Von Guericke.— TVaiM.

124

THE SCIENCE OF MECHANICS,

vacnam.

Quantita-
tive data.

The fall of which confirms in a simple manner the view of Galileo
that when the resistance of the air has been eliminated
light and heavy bodies both fall with the same velo-
city. In an exhausted glass tube a leaden bullet and a
piece of paper are placed. Putting the tube in a ver-
tical position and quickly turning it about a horizontal
axis through an angle of i8o^, both bodies will be seen
to arrive simultaneously at the bottom of the tube.

Of the quantitative data we will mention the fol-
lowing. The atmospheric pressure that supports a
column of mercury of 76 cm. is easily calculated from
the specific gravity 13*60 of mercury to be 1-0336 kg.
to I sq.cm. The weight of 1000 cu.cm. of pure, dry
air at 0° C. and 760 mm. of pressure at Paris at an ele-
vation of 6 metres will be found to be i -293 grams,
and the corresponding specific gravity, referred to
water, to be 0-001293.

Thediscov- 1 1. Guericke knew of only ^«^ kind of air. We

ery of other

gaseous may imagine therefore the excitement it created when

' in 1755 Black discovered carbonic acid gas (fixed air)

and Cavendish in 1766 hydrogen (inflammable air),

discoveries which were soon followed by other similar

ones. The dissimilar

physical properties of
gases are very strik-
ing. Faraday has il-
lustrated their great
inequality , of weight
by a beautiful lecture-
experiment. If from
a balance in equilib-
rium, we suspend (Fig. 84) two beakers A^ B, the one
in an upright position and the other with its opening
downwards, we may pour heavy carbonic acid gas from

Fig. 84.

THE PRINCIPLES OF STATICS. 125

above into the one and light hydrogen from beneath
into the other. In both instances the balance turns in
the direction of the arrow. To-day, as we know, the
decanting of gases can be made directly visible by the
optical method of Foucault and Toeppler.

12. Soon after Torricelli's discovery, attempts were The merco-
made to employ practically the vacuum thus produced, pump.
The so-called mercurial air-pumps were tried. But no

such instrument was successful until the present cen-
tury. The mercurial air-pumps now in common use
are really barometers of which the extremities are sup-
plied with large expansions and so connected that their
difference of level may be easily varied. The mercury
takes the place of the pistop of the ordinary air-pump.

13. The expansive force of the air, a property ob- Boyle's law.
served by Guericke, was more accurately investigated

by Boyle, and, later, by Mariotte. The law which
both found is as follows. If Fbe called the volume of
a given quantity of air and P its pressure on unit area
of the containing vessel, then the product V, P is
always = a constant quantity. If the volume of the
enclosed air be reduced one-half, the air will exert
double the pressure on unit of area ; if the volume of
the enclosed quantity be doubled, the pressure will
sink to one-half ; and so on. It is quite correct — as a
number of English writers have maintained in recent
times — that Boyle and not Mariotte is to be regarded
as the discoverer of the law that usually goes by
Mariotte's name. Not only is this true, but it must
also be added that Boyle knew that the law did not
hold exactly, whereas this fact appears to have escaped
Mariotte.

The method pursued by Mariotte in the ascertain-
ment of the law was very simple. He partially filled

126

THE SCIENCE OF MECHANICS,

His appa
ratus.

Mariotte's Torricellian tubes with mercury, measured the volume

expert-

ments. of the air remaining, and then performed the Torricel-
lian experiment. The new volume of
air was thus obtained, and by subtract-
ing the height of the column of mer-
cury from the barometer-height, also
the new pressure to which the same
quantity of air was now subjected.
To condense the air Mariotte em-
^^ ployed a siphon-tube with vertical

Fig. 85. arms. The smaller arm in which the

air was contained was sealed at the
upper end j the longer, into which the
mercury was poured, was open at the
y, upper end. The volume of the air
was read off on the graduated tube,
and to the difference of level of the
mercury in the two arms the barometer-
height was added. At the present day
both sets of experiments are performed
in the simplest manner by fastening a
cylindrical glass tube (Fig. 86) rr,
closed at the top, to a vertical scale
and connecting it by a caoutchouc
tube k k with a second open glass tube
r* r\ which is movable up and down
the scale. If the tubes be partly filled
with mercury, any difference of level
whatsoever of the two surfaces of mer-
Fig. 86. cury may be produced by displacing

r* r\ and the corresponding variations of volume of the
air enclosed in r r observed.

It struck Mariotte on the occasion of his investiga-
tions that any small quantity of air cut off completely

THE PRINCIPLES OF STATICS. 127

from the rest of the atmosphere and therefore notTheexpan-

sive force of

directly affected by the latter*s weight, also supported isolated

. . portions of

the barometer- colum n : as where, to give an instance, theatmos-

. , phere.

the open arm of a barometer- tube is closed. The simple
explanation of this phenomenon, which, of course,
Mariotte immediately found, is this, that the air before
enclosure must have been compressed to a point at
which its tension balanced the gravitational pressure
of the atmosphere ; that is to say, to a poini at which
it exerted an equivalent elastic pressure.

We shall not enter here into the details of the ar-
rangement and use of air-pumps, which are readily
understood from the law of Boyle and Mariotte.

14. It simply remains for us to remark, that the dis-
coveries of aerostatics furnished so much that was new
and wonderful that a valuable intellectual stimulus pro-
ceeded from the science.

CHAPTER II.

Dvni
whol

kamics
lolly a
modern
science.

THE DEVELOPMENT OF THE PRINCIPLES OF

DYNAMICS.

Galileo's achievements.

I. We now pass to the discussion of the funda-
mental principles of dynamics. This is entirely a mod-
ern science. The mechanical speculations of the an-
cients, particularly of the Greeks, related wholly to
statics. Dynamics was founded by Galileo. We shall
readily recognise the correctness of this assertion if we
but consider a moment a few propositions held by the
Aristotelians of Galileo's time. To explain the descent
of heavy bodies and the rising of light bodies, (in li-
quids for instance, ) it was assumed that every thing and
object sought its place : the place of heavy bodies was
below, the place of light bodies was above. Motions
were divided into natural motions, as that of descent,
and violent motions, as, for example, that of a pro-
jectile. From some few superficial experiments and
observations, philosophers had concluded that heavy
bodies fall more quickly and lighter bodies more slowly,
or, more precisely, that bodies of greater weight fall
more quickly and those of less weight more slowly. It
is sufficiently obvious from this that the dynamical
knowledge of the ancients, particularly of the Greeks,
was very insignificant, and that it was left to modern

THE PRINCIPLES OP DYNAMICS. 129

times to lay the true foundations of this department of

inquiry.

2. The treatise Discorsi e dimostrazioni maUmatiche,
in which Galileo communicated to the world the &rst

I30 THE SCIENCE OF MECHANICS,

Galileo's dynamical investigation of the laws of falling bodies,
tionof the appeared in 1638. The modern spirit that Galileo dis-

laws of fall-

ing bodies, covers is evidenced here, at the very outset, by the fact
that he does not ask why heavy bodies fall, but pro-
pounds the question, Hoiv do heavy bodies fall ? in
agreement with what law do freely falling bodies move?
The method he employs to ascertain this law is this.
He makes certain assumptions. He does not, however,
like Aristotle, rest there, but endeavors to ascertain by
trial whether they are correct or not.

His first. The first theory on which he lights is the following.

theory. * It seems in his eyes plausible that a freely falling body,
inasmuch as it is plain that its velocity is constantly
on the increase, so moves that its velocity is double
after traversing double the distance, and triple after
traversing triple the distance ; in short, that the veloci-
ties acquired in the descent increase proportionally
to the distances descended through. Before he pro-
ceeds to test experimentally this hypothesis, he reasons
on it logically, implicates himself, however, in so doing,
in a fallacy. He says, if a body has acquired a certain
velocity in the first distance descended through, double
the velocity in double such distance descended through,
and so on ; that is to say, if the velocity in the second
instance is double what it is in the first, then the double
distance will be traversed in the same time as the origi-
nal simple distance. If, accordingly, in the case of
the double distance we conceive the first half trav-
ersed, no time will, it would seem, fall to the account
of the second half. The motion of a falling body ap-
pears, therefore, to take place instantaneously ; which
not only contradicts the hypothesis but also ocular evi-
dence. We shall revert to this peculiar fallacy of
Galileo's later on.

THE PRINCIPLES OF DYNAMICS,

131

3. After Galileo fancied he had discovered this as- His second,
sumption to be untenable, he made a second one, ac- sumption.
cording to which the velocity acquired is proportional
to the time of the descent. That is, if a body fall once,
and then fall again during twice as long an interval of
time as it first fell, it will attain in the second instance
double the velocity it acquired in the first. He found
no self-contradiction in this theory, and he accordingly
proceeded to investigate by experiment whether the
assumption accorded with observed facts. It was dif-
ficult to prove by any direct means that the velocity
acquired was proportional to the time of descent. It
was easier, however, to investigate by what law the
distance increased with the time ; and he consequently
deduced from his assumption the relation that obtained
between the distance and the time, and tested this by
experiment. The deduction ^

is simple, distinct, and per- ^^

fectly correct. He draws
(Fig. 87) a straight line, and
on it cuts off successive por- Ca-
tions that represent to him Fig. 87.
the times elapsed. At the extremities of these por-
tions he erects perpendiculars (ordinates), and these
represent the velocities acquired. Any portion OGoi
the line OA denotes, therefore, the time of descent
elapsed, and the corresponding perpendicular GIf the
velocity acquired in such time.

If, now, we fix our attention on the progress of the
velocities, we shall observe with Galileo the following
fact : namely, that at the instant C, at which one-half
OC of the time of descent OA has elapsed, the velocity
CD is also one-half of the final velocity AB.

If now we examine two instants of time, £ and G,

Discassion
and eluci-
dation of
the true
theory.

132 THE SCIENCE OF MECHANICS.

Uniformly equally distant in opposite directions from the instant

accelerated

motion. C, we shall observe that the velocity HG exceeds the
mean velocity CD by the same amount that EF falls
short of it. For every instant antecedent to C there
exists a corresponding one equally distant from it sub-
sequent to C, Whatever loss, therefore, as compared
with uniform motion with half the final velocity, is suf-
fered in the first half of the motion, such loss is made
up in the second half. The distance fallen through we
may consequently regard as having been uniformly de-
scribed with half the final velocity. If, accordingly,
we make the final velocity v proportional to the time
of descent /, we shall obtain 7* = gt, where g denotes
the final velocity acquired in unit of time — the so-called
acceleration. The space s descended through is there-
fore given by the equation s = (^//2) / or s =zg/^/2.
Motion of this sort, in which, agreeably to the assump-
tion, equal velocities constantly accrue in equal inter-
vals of time, we call uniformly accelerated motion.
Table of the If we collect the times of descent, the final veloci-
iocities.and ties, and the distances traversed, we shall obtain the

distances of ^ , , , ,

descent, followmg table :

1^-

1 X 1 . f

2g.

2X2.'^-

3^.

3 X 3 . 1

4^.

4X4. f

•
•
•

•
•
«

fg-

'X'l

THE PRINCIPLES OF DYNAMICS. 133

4. The. relation obtaining between / and s admits Experimen-

t&l verifies*

of experimental proof ; and this Galileo accomplished tion of the
in the manner which we shall now describe.

We must first remark that no part of the knowledge
and ideas on this subject with which we are now so
familiar, existed in Galileo's time^ but that Galileo had
to create these ideas and means for us. Accordingly,
it was impossible for him to proceed as we should do
to-day, and he was obliged, therefore, to pursue a dif-
ferent method. He first sought to retard the motion
of descent, that it might be more accurately observed.
He made observations on balls, which he caused to
roll down inclined planes (grooves); assuming that only
the velocity of the motion would be lessened here, but
that the form of the law of desert would remain un-
modified. If, beginning from the upper extremity, the Jhe arti-
fices em-
distances I, 4, 9, 16 ... be notched off on the groove, ployed.

the respective times of descent will be representable,
it was assumed, by the numbers i, 2, 3, 4 . . . ; a result
which was, be it added, confirmed. The observation of
the times involved, Galileo accomplished in a very in-
genious manner. There were no clocks of the modern
kind in his day : such were first rendered possible by
the dynamical knowledge of which Galileo laid the
foundations. The mechanical clocks which were used
were very inaccurate, and were available only for the
measurement of great spaces of time. Moreover, it
was chiefly water-clocks and sand-glasses that were in
use — in the form in which they had been handed down
from the ancients. Galileo, now, constructed a very
simple clock of this kind, which he especially adjusted
to the measurement of small spaces of time ; a thing
not customary in those days. It consisted of a vessel of
water of very large transverse dimensions, having in

134 TW:^ SCIENCE OF MECHANICS,

Galileo's the bottom a minute orifice which was closed with the

clock.

finger. As soon as the ball began to roll down the in-
clined plane Galileo removed his finger and allowed the
water to flow out on a balance ; when the ball had ar-
rived at the terminus of its path he closed the orifice.
As the pressure-height of the fluid did not, owing to
the great transverse dimensions of the vessel, percept-
ibly change, the weights of the water discharged from
the orifice were proportional to the times. It was in
this way actually shown that the times increased simply,
while the spaces fallen through increased quadratically.
The inference from Galileo's assumption was thus con-
firmed by experiment, and with it the assumption itself.
The reia- c. To form some notion of the relation which sub-

tion of mo- ,

tion on an sists between motiqp on an inclined plane and that of

inclined .

plane to free descent, Galileo made the assumption, that a body
descent, which falls through the height of an inclined plane
attains the same final velocity as a body which falls
through its length. This is an assumption that will
strike us as rather a bold one ; but in the manner in
which it was enunciated and employed by Galileo, it is
quite natural. We shall endeavor to explain the way by
which he was led to it. He says : If a body fall freely
downwards, its velocity increases proportionally to the
time. When, then, the body has arrived at a point be-
low, let us imagine its velocity reversed and directed
upwards ; the body then, it is clear, will rise. We make
the observation that its motion in this case is a reflection,
so to speak, of its motion in the first case. As then its
velocity increased proportionally to the time of descent,
it will now, conversely, diminish in that proportion.
When the body has continued to rise for as long a
time as it descended, and has reached the height from
which it originally fell, its velocity will be reduced to

THE PRINCIPLES OF DYNAMICS, 135

zero. We perceive, therefore, that a body will rise, justifica-
in virtue of the velocity acquired in its descent, just as assumption
high as it has fallen. If, accordingly, a body falling final veioc-
down an inclined plane could acquire a velocity which motions are
would enable it, when placed on a differently inclined
plane, to rise higher than the point from which it had
fallen, we should be able to effect the elevation of
bodies by gravity alone. There is contained, accord-
ingly, in this assumption, that the velocity acquired by
a body in descent depends solely on the vertical height
fallen through and is independent of the inclination of
the path, nothing more than the uncontradictory ap-
prehension and recognition of the/ar/ that heavy bodies
do not possess the tendency to rise, but only the ten-
dency to fall. If we should assume that a body fall-
ing down the length of an inclined plane in some way
or other attained a greater velocity than a body that
fell through its height, we should only have to let the
body pass with the acquired velocity to another in-
clined or vertical plane to make it rise to a greater ver-
tical height than it had fallen from. And if the velo-
city attained on the inclined plane were less, we should
only have to reverse the process to obtain the same re-
sult. In both instances a heavy body could, by an ap-
propriate arrangement of inclined planes, be forced
continually upwards solely by its own weight — a state
of things which wholly contradicts our instinctive
knowledge of the nature of heavy bodies.

6. Galileo, in this case, again, did not stop with
the mere philosophical and logical discussion of his
assumption, but tested it by comparison with expe-
rience.

He took a simple filar pendulum (Fig. 88) with a
heavy ball attached. Lifting the pendulum, while

136

THE SC/E^/CE OF MECHANICS,

Galileo's elongated its full length, to the level of a given altitude,

tai verifica- and then letting it fall, it ascended to the same level

assumption on the Opposite side. If it does not do so exactly ^

Galileo said, the resistance of the air must be the cause

of the deficit. This is inferrible from the fact that the

deficiency is greater in the case of a cork ball than it is

Effected by in the case of a heavy metal one. However, this neg-
fmpeding lected, the body ascends to the same altitude on the
of apendu- Opposite side. Now it is permissible to regard the mo-
nm s nng. ^.^^ ^^ ^ pendulum in the arc of a circle as a motion

of descent along a series of inclined planes of different
inclinations. This seen, we can, with Galileo, easily
cause the body to rise on a different arc — on a different
series of inclined planes. This we accomplish by driv-
ing in at one side of the thread, as it vertically hangs,
a nail / or ^, which will prevent any given portion of
the thread from taking part in the second half of the
motion. The moment the thread arrives at the line of
equilibrium and strikes the nail, the ball, which has
fallen through ba, will begin to ascend by a different
series of inclined planes, and describe the arc am ox an.
Now if the inclination of the planes had any influence

THE PRINCIPLES OF D YNAMICS, 137

on the velocity of descent, the body could not rise to
the same horizontal level from which it had fallen.
But it does. By driving the nail sufficiently low down,
we may shorten the pendulum for half of an oscillation
as much as we please ; the phenomenon, however, al-
ways remains the same. If the nail h be driven so low
down that the remainder of the string cannot reach to
the plane E^ the ball will turn completely over and
wind the thread round the nail ; because when it has
attained the greatest height it can reach it still has a
residual velocity left,

7. If we assume thus, that the same final velocity is The as-
sumption

attained on an inclined plane whether the body fall leads to the

... »*w of rela-

through the height or the length of the plane, — in which tive accei-
assumption nothing more is contained than that a body sought.
rises by virtue of the velocity it has acquired in falling
just as high as it has fallen, — we shall easily arrive,
with Galileo, at the perception that the times of the de-
scent along the height and the length of an inclined
plane are in the simple proportion of the height and
the length ; or, what is the same, that the accelerations
are inversely proportional to the times of descent.
The acceleration along the height will consequently
bear to the acceleration along ^^
the length the proportion of the
length to the height. Let AB
(Fig. 89) be the height and AC^
the length of the inclined plane. Fig. 89.

Both will be descended through in uniformly accel-
erated motion in the times / and /^ with the final ve-
locity V, Therefore,

138

THE SCIENCE OF MECHANICS,

If the accelerations along the height and the length be
called respectively g and g^y we also have

v = gi and v^=^ g. /, , whence -^- = - = - -- = sin a.

^ ^ g t^ AC

In this way we are able to deduce from the accel-
eration on an inclined plane the acceleration of free
descent.
A corollary From this proposition Galileo deduces several cor-

of the pre- . * '^

ceding law. ollaries, some of which have passed into our elementary
text-books. The accelerations along the height and
length are in the inverse proportion of the height and
length. If now we cause one body to fall along the
length of an inclined plane and simultaneously another
to fall freely along its height, and ask what the dis-
tances are that are traversed by the two in equal inter-
vals of time, the solution of the problem will be readily
found (Fig. 90) by simply letting fall from B a perpen-
dicular on the length. The part AD, thus cut off, will
be the distance traversed by the one body on the in-
clined plane, while the second body is freely falling

through the height of the plane.

Fig. 90. Fig. 91.

Relative If we describe (Fig. 91) a circle on AB as diame-

scription of ter, the circle will pass through Z>, because Z> is a
anddiame- right angle. It wiU be seen thus, that we can imagine
cies. any number of inclined planes, AE, AF, of any degree

of inclination, passing through A, and that in every

THE PRINCIPLES OF D YNAMICS,

139

circles.

case the chords A Gy AH drawn in this circle from the
upper extremity of the diameter will be traversed in
the same time by a falling body as the vertical diame-
ter itself. Since, obviously, only the lengths and in-
clinations are essential here, we may also draw the
chords in question from the lower extremity of the
diameter, and say generally : The vertical diameter
of a circle is described by a falling particle in the same
time that any chord through either extremity is so
described.

We shall present another corollary, which, in the The figures
pretty form in which Galileo gave it, is usually no bodies fill-
longer incorporated in elementary expositions. We chords of
imagine gutters radiating in a vertical plane from a
common point ^ at a
number of different
degrees of inclination
to the horizon (Fig.
92). We place at their
common extremity A
a like number of heavy
bodies and cause them
to begin simultaneous-
ly their motion of des-
cent. The bodies will
always form at any one

instant of time a circle. After the lapse of a longer time
they will be found in a circle of larger radius, and the
radii increase proportionally to the squares of the
times. If we imagine the gutters to radiate in a space
instead of a plane, the falling bodies will always form
a sphere, and the radii of the spheres will increase pro-
portionally to the squares of the times. This will be

140

THE SCIENCE OF MECHANICS.

ti.

perceived by imagining the figure revolved about
perpendicular A V.
Character 8. We see tlius, — as deserves again to be briefly

inquiries, noticed, — that Galileo did not supply us with a theory
of the falling of bodies, but investigated and estab-
lished, wholly without preformed opinions, the actual
facts of falling.

Gradually adapting, on this occasion, his thoughts
to the facts, and everywhere logically abiding by the
ideas he had reached, he hit on a conception, which to
himself, perhaps less than to his successors, appeared
in tne light of a new law. In all his reasonings, Galileo
followed, to the greatest advantage of science, a prin-
ciple which might appropriately be called the principle
of continuity. Once we have reached a theory that ap-
plies to a particular case, we proceed gradually to
modify in thought the conditions of that case, as far
as it is at all possible, and endeavor in so doing to
adhere throughout as closely as we can to the concep-
tion originally reached. There is no method of pro-
cedure more surely calculated to lead to that compre-
hension of all natural phenomena which is the simplest
and also attainable with the least expenditure of men-
tality and feeling. (Compare Appendix, I.)

A particular instance will show more clearly than
any general remarks what we mean. Galileo con-

The prin-
ciple of
continuity.

Fig. 93-

siders (Fig. 93) a body which is falling down the in-
clined plane AB, and which, being placed with the

THE PRINCIPLES OF DYNAMICS, 141

velocity thus acquired on a second plane BC, for ex- Galileo's

discovery

ample, ascends this second plane. On all planes BC, of the so-

called law

BDy and so forth, it ascends to the horizontal plane of inertia,
that passes through A, But, just as it falls on BD
with less acceleration than it does on BC, so similarly
it will ascend on BD with less retardation than it will
on BC. The nearer the planes BCy BD, BE, BF ap-
proach to the horizontal plane BH, the less will the
retardation of the body on those planes be, and the
longer and further will it move on them. On the hori-
zontal plane BH the retardation vanishes entirely (that
is, of course, neglecting friction and the resistance of
the air), and the body will continue to move infinitely
long and infinitely far with constant velocity. Thus ad-
vancing to the limiting case of the problem presented,
Galileo discovers the so-called law of inertia, according
to which a body not under the influence of forces, i. e.
of special circumstances that change motion, will re-
tain forever its velocity (and direction). We shall
presently revert to this subject.

9. The motion of falling that Galileo found actually The deduc-
\o exist, is, accordingly, a motion of which the velocity idea of nni-
increases proportionally to the time — a so-called uni- ccierated
formly accelerated motion.

It would be an anachronism and utterly unhistorical
to attempt, as is sometimes done, to derive the uniformly
accelerated motion of falling bodies from the constant
action of the force of gravity. ** Gravity is a constant
force ; consequently it generates in equal elements of
time equal increments of velocity ; thus, the motion
produced is uniformly accelerated." Any exposition
such as this would be unhistorical, and would put the
whole discovery in a false light, for the reason that the
notion of force as we hold it to-day was first created

142 THE SCIENCE OF MECHANICS,

Forces and bv Galileo. Before Galileo force was known solely as
tions. pressure. Now, no one can know, who has not learned
it from experience, that generally pressure produces
motion, much less in what manner pressure passes into
motion ; that not position, nor velocity, but accelera-
tion, is determined by it. This cannot be philosophi-
cally deduced from the conception, itself. Conjectures
may be set up concerning it. But experience alone can
definitively inform us with regard to it.

10. It is not by any means self-evident, therefore,
that the circumstances which determine motion, that
is, forces, immediately produce accelerations. A glance
at other departments of physics will at once make this
clear. The differences of temperature of bodies also
determine alterations. However, by differences of tem-
perature not compensatory accelerations are deter-
mined, but compensatory velocities.

The fact That it is accelerations which are the immediate ef-

determine fects of the circumstances that determine motion, that
tions is an is, of the forces, is a fact which Gsilileo perceived in the
toPfact*" natural phenomena. Others before him had also per-
ceived many things. The assertion that everything seeks
its place also involves a correct observation. The ob-
servation, however, does not hold good in all cases,
and it is not exhaustive. If we cast a stone into the
air, for example, it no longer seeks its place ; since its
place is below. But the acceleration towards the earth,
the retardation of the upward motion, the fact that Ga-
lileo perceived, is still present. His observation always
remains correct ; it holds true more generally ; it em-
braces in one mental effort mucA more,

11. We have already remarked that Galileo dis-
covered the so-called law of inertia quite incidentally.
A body on which, as we are wont to say, a force acts.

THE PRINCIPLES OF DYNAMICS. 143

preserves its direction and velocity unchanged. The History of
fortunes of this law of inertia have been strange. It called law
appears never to have played a prominent part in Gali-
leo's thought. But Galileo's successors, particularly
Huygens and Newton, formulated it as an independent
law. Nay, some have even made of inertia a general
property of matter. We shall readily perceive, how-
ever, that the law of inertia is not at all an indepen-
dent law, but is contained implicitly in Galileo's per-
ception that all circumstances determinative of motion,
or forces, produce accelerations.

In fact, if a force determine, not position, not velo- The law a
city, but acceleration, change of velocity, it stands to ference
reason that where there is no force there will be no leo's funda

...... . mental ob-

change of velocity. It is not necessary to enunciate aervation.
this in independent form. The embarrassment of the
neophyte, which also overcame the great investigators
in the face of the great mass of new material presented,

■

alone could have led them to conceive the same fact as
two different facts and to formulate it twice.

In any event, to represent inertia as self-evident, or Erroneous
to derive it from the general proposition that ''the ef- deducing u
feet of a cause persists," is totally wrong. Only a
mistaken straining after rigid logic can lead us so out
of the way. Nothing is to be accomplished in the pres-
ent domain with scholastic propositions like the one
just cited. We may easily convince ourselves that the
contrary proposition, "cessante causa cessat effectus,"
is as well supported by reason. If we call the acquired
velocity "the effect," then the first proposition is cor-
rect ; if we call the acceleration "effect," then the sec-
ond proposition holds.

1 2. We shall now examine Galileo's researches from
another side. He began his investigations with the

144

THE SCIENCE OF MECHANICS,

Notion of notions familiar to his time — notions developed mainly
it existed in in the practical arts. One notion of this kind was that
time. of velocity, which is very readily obtained from the con-
sideration of a uniform motion. If a body traverse in
every second of time the same distance ^, the distance
traversed at the end of / seconds will be j = r/. The
distance c traversed in a second of time we call the ve-
locity, and obtain it from the examination of any por-
tion of the distance and the corresponding time by the
help of the equation c = j//, that is, by dividing the
number which is the measure of the distance traversed
by the number which is the measure of the time elapsed.
Now, Galileo could not complete his investigations
without tacitly modifying and extending the traditional
idea of velocity. Let us represent for distinctness sake

in I (Fig. 94) a uniform motion, in 2 a variable motion,
by laying off as abscissae in the direction OA the elapsed
times, and erecting as ordinates in the direction AB the
distances traversed. Now, in i, whatever increment
of the distance we may divide by the corresponding in-
crement of the time, in all cases we obtain for the ve-
locity c the same value. But if we were thus to proceed
in 2, we should obtain widely differing values, and
therefore the word "velocity " as ordinarily understood,
ceases in this case to be unequivocal. If, however, we
consider the incr^asQ of the distance in a sufficiently

THE PRINCIPLES OF D YNAMICS. 145

small element of time, where the element of the curve Galileo's
m 2 approaches to a straight line, we may regard the tion of this
increase as uniform. The velocity in this element of
the motion we may then define as the quotient, J s/A /,
of the element of the time into the corresponding ele-
ment of the distance. Still more precisely, the velocity
at any instant is defined as the limiting value which
the ratio A sjA t assumes as the elements become in-
finitely small — a value designated by dsjdt. This new
notion includes the old one as a particular case, and is,
moreover, immediately applicable to uniform motion.
Although the express formulation of this idea, as thus
extended, did not take place till long after Galileo, we
see none the less that he made use of it in his reason-
ings.

13. An entirely new notion to which Galileo was The notion
led is the idea of acceleration. In uniformly acceler- tion.
ated motion the velocities increase with the time
agreeably to the same law as in uniform motion the
spaces increase with the times. If we call v the velo-
city acquired in time /, then v = gt. Here g denotes
the increment of the velocity in unit of time or the ac-
celera'tion, which we also obtain from the equation
g = v/t. When the investigation of variably accel-
erated motions was begun, this notion of accelera-
tion had to experience an extension similar to that of
the notion of velocity. If in i and 2 the times be again
drawn as abscissae, but now the velocities as ordinates,
we may go through anew the whole train of the pre-
ceding reasoning and define the acceleration as dvjdt^
where dv denotes an infinitely small increment of the
velocity and dt the corresponding increment of the
time. In the notation of the differential calculus we

146

THE SCIENCE OF MECHANICS,

have for the acceleration of a rectilinear motion, qt =
Graphic The ideas here developed are susceptible, moreover,

representa-

rion of of graphic representation. If we lay off the times as

these ideas. .

abscissae and the distances as ordinates, we shall per-
ceive, that the velocity at each instant is measured by
the slope of the curve of the distance. If in a similar
manner we put times and velocities together, we shall
see that the acceleration of the instant is measured by
the slope of the curve of the velocity. The course of
the latter slope is, indeed, also capable of being traced
in the curve of distances, as will be perceived from
the following considerations. Let us imagine, in the

Fig. 95-

Fig. 96.

Thecurve usual manner (Fig. 95), a uniform motion represented

of distance. \ ^ ^^n r

by a straight Ime OCD. Let us compare with this a
motion OCE the velocity of which in the second half
of the time is greater, and another motion OCF of
which the velocity is in the same proportion smaller.
In the first case, accordingly, we shall have to erect for
the time OB = 2 OA, an ordinate greater than BZ> ==
2 AC; in the second case, an ordinate less than BD,
We see thus, without difficulty, that a curve of dis-
tance convex to the axis of the time-abscissae corre-
sponds to accelerated motion, and a curve concave
thereto to retarded motion. If we imagine a lead-pen-
cil to perform a vertical motion of any kind and in

THE PRINCIPLES OF DYNAMICS. 147

front of it during its motion a piece of paper to be uni-
formly drawn along from right to left and the pencil to
thus execute the drawing in Fig. 96, we shall be able to
read off from the drawing the peculiarities of the mo-
tion. At a the velocity of the pencil was directed up-
wardsy at b it was greater, at c it was =0, at </ it was
directed downwards, at e it was again = 0. At a^ b^
dy e, the acceleration was directed upwards, at c down-
wards ; at c and e it was greatest.

14. The summary representation of what Galileo Tabular
discovered is best made by a table of times, acquired mem of ca-

^ ^ lileo'8dis-

/. V, S. "'^^

»f

■ • ■ •

velocities, and traversed distances. But the numbers The table

ooav bo re~

follow so simple a law, — one immediately recognisable, placed by,
— that there is nothing to prevent our replacing the constnic-
table by a rule for its construction. If we examine the
relation that connects the first and second columns, we
shall find that it is expressed by the equation v = gt,
which, in its last analysis, is nothing but an abbrevi-
ated direction for constructing the first two columns
of the table. The relation connecting the first and third
columns is given by the equation s =.gt^ jo.. The con-
nection of the second and third columns is represented
by s=^v^/2g.

148 THE SCIENCE OF MECHANICS,

The rules. Of the three relations

v = gt

rr/2
i" = -

strictly, the first two only were employed by Galileo.
Huygens was the first who evinced a higher apprecia-
tion of the third, and laid, in thus doing, the founda-
tions of important advances.
A remark 1 5. We may add a remark in connection with

tionofthe this table that is very valuable. It has been stated
t^elSies. previously that a body, by virtue of the velocity it has
acquired in its fall, is able to rise again to its origi-
nal height, in doing which its velocity diminishes in
the same way (with respect to time and space) as it
increased in falling. Now a freely falling body ac-
quires in double time of descent double velocity, but
falls in this double time through four times the simple
distance. A body, therefore, to which we impart a ver-
tically upward double velocity will ascend twice as
long a time, hxiXfour times as high as a body to which
the simple velocity has been imparted.
The dispute It was remarked, very soon after Galileo, that there
tesians and is inherent in the velocity of a body a something that
ians on the corresponds to a force — a something, that is, by which

measure of . , • rr •! • 1

force. a force can be overcome, a certam " efficacy, as it has
been aptly termed. The only point that was debated
was, whether this efficacy was to be reckoned propor-
tional to the velocity or to the square of the velocity.
The Cartesians held the former, the Leibnitzians the
latter. But it will be perceived that the question in-
volves no dispute whatever. The body with the double
velocity overcomes a given force through double the

THE PRINCIPLES OF D YNAMICS. 149

time, but through four times the distance. With re-
spect to time, therefore, its efficacy is proportional to
the velocity ; with respect to distance, to the square of
the velocity. D'Alembert drew attention to this mis-
understanding, although in not very distinct terms. It
is to be especially remarked, however, that Huygens's
thoughts on this question were perfectly clear.

16. The experimental procedure by which, at the The present

1 J* -n J • experimen-

present day, the laws of falling bodies are verified, istaimeansof

r 1*1 T« V J venfyinj?

somewhat different from that of Galileo. Two methods the jawa of
may be employed. Either the motion of falling, which ies.
from its rapidity is difficult to observe directly, is so
retarded, without altering the law, as to be easily ob-
served ; or the motion of falling is not altered at all,
but our means of observation are improved in deli-
cacy. On the first principle Galileo's inclined
gutter and At wood's machine rest. Atwood's
machine consists (Fig. 97) of an easily run-
ning pulley, over which is thrown a thread,
to whose extremities two equal weights P are
attached. If upon one of the weights P we
lay a third small weight p^ a uniformly accel- r\P
erated motion will be set up by the over- p>k-97.
weight, having the acceleration {p/'iP-^- f) g — a result
that will be readily obtained when we shall have dis-
cussed the notion of **mass." Now by means of a
graduated vertical standard connected with the pulley
it may easily be shown that in the times i, 2, 3,. 4 ... .
the distances i, 4, 9, 16. . . . are traversed. The final
velocity corresponding to any given time of descent is
investigated by catching the small additional weight,/,
which is shaped so as to project beyond the outline of
P, in a ring through which the falling body passes,
after which the motion continues without acceleration.

150 THE SCIENCE OF MECHANICS.

The appa- The apparatus of Morin is based on a different prin-
Morin, La- ciple. A body to which a writing pencil is attached
pich, and describes on a vertical sheet of paper, which is drawn
uniformly across it by a clock-work, a horizontal straight
line. If the body fall while the paper is not in motion,
it will describe a vertical straight line. If the two
motions are combined, a parabola will be produced,
of which the horizontal abscissae correspond to the
elapsed times and the vertical ordinates to the dis-
tances of descent described. For the abscissse i, 2,
3, 4 ... . we obtain the ordinates i, 4, 9, 16 ... . By
an unessential modification, Morin employed instead of
a plane sheet of paper, a rapidly rotating cylindrical
drum with vertical axis, by the side of which the body
fell down a guiding wire. A different apparatus, based
on the same principle, was invented, independently, by
Laborde, Lippich, and Von Babo. A lampblacked
sheet of glass (Fig. 98^;) falls freely, while a horizon-
tally vibrating vertical rod, which in its first transit
through the position of equilibrium starts the motion
of descent, traces, by means of a quill, a curve on the
lampblacked surface. Owing to the constancy of the
period of vibration of the rod combined with the in-
creasing velocity of the descent, the undulations traced
by the rod became longer and longer. Thus (Fig. 98)
bc=i^ab, cd=i$aif, de=^']aby and so forth. The
law of falling bodies is clearly exhibited by this, since
ab -\- cb =z ^abj ab -^ be -\- cif = gab, and so forth.
The law of the velocity is confirmed by the inclinations
of the tangents at the points a, b, r, //, and so forth. If
the time of oscillation of the rod be known, the value
of g is determinable from an experiment of this kind
with considerable exactness.

Wheatstone employed for the measurement of mi-

THE PRINCIPLES OF DYNAMICS. iji

□ute portions of time a rapidly operating clock-work Tbeds-
called a chronoscope, which is set in motion at the be- whm-
ginning of the time to be measured and stopped at the Hipp. ,
termination of it. Hipp has advantageously modified

Fig.9St

this method by simply causing a light index-hand to
be thrown by means of a clutch in and out of gear with
a rapidly moving wheel-work regulated by a vibrating
reed of steel tuned to a high note, and acting as an es-

152

THE SCIENCE OF MECHANICS,

Galileo's
minor in-
vestiga-
tions.

capement. The throwing in and out of gear is effected
by an electric current. Now if, as soon as the body be-
gins to fall, the current be interrupted, that is the hand
thrown into gear, and as soon as the body strikes the
platform below the current is closed, that is the hand
thrown out of gear, we can read by the distance the
index-hand has travelled the time of descent.

17. Among the further achievements of Galileo we
have yet to mention his ideas concerning the motion
of the pendulum, and his refutation of the view that
bodies of greater weight fall faster than bodies of less
weight. We shall revert to both of these points on an-
other occasion. It may be stated here, however, that
Galileo, on discovering the constancy of the period of
pendulum-oscillations, at once applied the pendulum
to pulse-measurements at the sick-bed, as well as pro-
posed its use in astronomical observations and to a cer-
tain extent employed it therein himself.

18. Of still greater importance are his investiga-

of projec- ,

tiles. tions concerning the motion of projectiles. A free body,

according to Galileo's view, constantly experiences a
vertical acceleration g towards the earth. If at the
beginning of its motion it is affected with a vertical

velocity c, its velocity at the
end of the time / will be z^ =
c -\- gt. An initial velocity up-
wards would have to be reck-
oned negative here. The dis-
tance described at the end of
p»R 99. time / is represented by the

equation s=^a-\- ct -\ ^gt^y where ct and ^gt^ are the
portions of the traversed distance that correspond re-
spectively to the uniform and the uniformly accelerated
motion. The constant a is to be put == 0 when we reckon

The motion

THE PRINCIPLES OF DYNAMICS, 153

the distance from the point that the body passes at time
/ = 0. When Galileo had once reached his fundamental
conception of dynamics, he easily recognised the case
of horizontal projection as a combination of two inde-
pendent motions, a horizontal uniform motion, and a
vertical uniformly accelerated motion. He thus intro-
duced into use the principle of the parallelogram of mo-
tions. Even oblique projection no longer presented the
slightest difficulty.

If a body receives a horizontal velocity c, it de- The carve
scribes in the horizontal direction in time / the distance tion a par-
y =:. ctf while simultaneously it falls in a vertical direc-
tion the distance jtr=^/2^2. Different motion-deter-
minative circumstances exercise no mutual effect on one
another, and the motions determined by them take
place independently of each other, Galileo was led to
this assumption by the attentive observation of the
phenomena ; and the assumption proved itself true.

For the curve which a body describes when the two
motions in question are compounded, we find, by em-
ploying the two equations above given, the expression
y =:V\2c ^ /g) X, It is the parabola of ApoUonius hav-
ing its parameter equal to c^/gond its axis vertical,
as Galileo knew.

We readily perceive with Galileo, that oblique pro- oblique
jection involves nothing new. The velocity c imparted
to a body at the angle a with the horizon is resolvable
into the horizontal component c . cos a and the vertical
component c . sin a. With the latter velocity thfe body
ascends during the same interval of time / which it
would take to acquire this velocity in falling vertically
downwards. Therefore, ^.sina=^/. When it has
reached its greatest height the vertical component of
its initial velocity has vanished, and from the point S

154 ^^^ SCIENCE OF MECHANICS.

onward (Fig. loo) it continues its motion as a horizon-
tal projection. If we examine any two epochs equally
distant in time, before and after the transit through 5,

we shall see that the body at
these two epochs is equally
distant from the perpendicu
lar through 5 and situated the
same distance below the hori-
zontal line through 5. The
Fig. loo. curve is therefore symmet-

rical with respect to the vertical line through 5. It
is a parabola with vertical axis and the parameter
(r cosflr)*/^.
The range To find the so-called range of projection, we have
o^projec- gjjjipiy to consider the horizontal motion during the
time of the rising and falling of the body. For the ascent
this time is, according to the equations above given,
/ = ^ sin ajgy and the same for the descent. With the
horizontal velocity c . cos a, therefore, the distance is

traversed

c sin oi c^ c^

w =^ c cos a . 2 = — 2 sin a cos a = sin 2 a,

g g g

The range of projection is greatest accordingly
when a = 45°, and equally great for any two angles

The mutual 1 9. The recognition of the Tdutwdl independence oi
dence of the forccs, or motion -determinative circumstances oc-

forces. . . ^ 1 • 1

currmg m nature, which was
reached and found expression
in the investigations relating to
projection, is important. A body
Fig. lox. jj^^y move (Fig. loi) in the di-

rection AB, while the space in which this motion oc-
curs is displaced in the direction A C. The body then

THE PRINCIPLES OF D YNAMICS. 155

goes from A to D. Now, this also happens if the two
circumstances that simultaneously determine the mo-
tions AB and A C, have no influence on one another.
It is easy to see that we may compound by the paral-
lelogram not only displacements that have taken place
but also velocities and accelerations that simultane-
ously take place.

II.

THE ACHIEVEMENTS OF HUVGENS.

I. The next in succession of the great mechanical in- Hamns's
quirers is Huvgens, who in every respect must beas\/in-
ranked as Galileo's peer. If, perhaps, his philosophical *'^*"'^'
endowments were less splendid than those of Galileo,
this deficiency was compensated for by the superiority
of his geometrical powers. Huygens not only continued
the researches which Galileo had begun, but he also
solved the first problems in the dynamics of several
masses^ whereas Galileo had throughout restricted him-
self to the dynamics of a single body.

The plenitude of Huygens's achievements is best Enumera-
seen in his HorologiumOscillaioriuniy which appeared in gens's
1673. The most important subjects there treated of formems.
the first time, are : the theory of the centre of oscilla-
tion, the invention and construction of the pendulum-
clock, the invention of the escapement, the determina-
tion of the acceleration of gravity, gy by pendulum-
observations, a proposition regarding the employment
of the length of the seconds pendulum as the unit of
length, the theorems respecting centrifugal force, the
mechanical and geometrical properties of cycloids, the
doctrine of evolutes, and the theory of the circle of
curvature.

156 THE SCIENCE OF MECHANICS.

2. With respect to the form of presentation of his
work, it is to be remarked that Huygens shares with

Galileo, in all its perfection, the latter's exalted and
inimitable candor. He is frank without reserve in the
presentment of the methods that led him to his dis-

THE PRINCIPLES OF D YA'AMICS.

coveries, and thus always
conducts his reader into the
full comprehension of his
performances. Nor had he
cause to conceal these
methods. If, some thou-
sand years hence, it will be
found that he was a man, it
will likewise be seen what
manner of man he was.
In our discussion of the
achievements of Huygens,
however, we shall have to
proceed in a somewhat dif-
ferent manner from that
which we pursued in the
case of Galileo. Galileo's
views, in their classical sim-
plicity, could be given in an
almost unmodified form.
With Huygens this is not
possible. The latter deals
with more complicated
problems; his mathematical
methods and notations be-
come inadequate and cum-
brous. For reasons of brev-
ity, therefore, we shall re-
produce all the conceptions
of which we treat, in mod-
em form, retaining, how-
ever, Huygens's essential
and characteristic ideas.

'i Psadulnin Clock.

X58 THE SCIENCE OF MECHANICS,

CentrifuKai 3. We begin with the investigations concerning
petal force, centrifugal force. When once we have recognised with
Galileo that force determines acceleration, we are im-
pelled, unavoidably, to ascribe every change of velocity
and consequently also every change in the direction of
a motion (since the direction is determined by three
velocity-components perpendicular to one another) to
a force. If, therefore, any body attached to a string,
say a stone, is swung uniformly round in a circle, the
curvilinear motion which it performs is intelligible only
on the supposition of a constant force that deflects the
body from the rectilinear path. The tension of the
string is this force ; by it the body is constantly deflected
from the rectilinear path and made to move towards
the centre of the circle. This tension, accordingly, rep-
resents a centripetal force. On the other hand, the axis
also, or the fixed centre, is acted on by the tension of
the string, and in this aspect the tension of the string
appears as a centrifugal force.

iQ-

Fig. xoa. Fig. 103.

Let US suppose that we have a body to which a ve-
locity has been imparted and which is maintained in
uniform motion in a circle by an acceleration constantly
directed towards the centre. The conditions on which
this acceleration depends, it is our purpose to investi-
gate. We imagine (Fig. 102) two equal circles uni-

THE PRINCIPLES OF D YNAMICS. 159

formly travelled round by two bodies ; the velocities in uniform
the circles I and II bear to each other the proportion e^uai

circles

1:2. If in the two circles we consider any same arc-
element corresponding to some very small angle at, then
the corresponding element s of the distance that the
bodies in consequence of the centripetal acceleration
have departed from the rectilinear path (the tangent),
will also be the same. If we call q)^ and q)^ the re-
spective accelerations, and r and r/2 the time-elements
for the angle or, we find by Galileo's law

<Py = :^y 92 = ^ -^» ^^^^ 's to say (p^ = 4(p^,

Therefore, by generalisation, in equal circles the
centripetal acceleration is proportional to the square of
the velocity of the motion.

Let us now consider the motion in the circles I and Uniform

xx-r-i' 1 fri't 1 \ molion in

II (Fig. 103), the radii of which are to each other as unequal
I : 2, and let us take for the ratio of the velocities of
the motions also 1:2, so that like arc-elements are
travelled through in equal times, (p^, (p^, s, is denote
the accelerations and the elements of the distance trav-
ersed ; r is the element of the time, equal for both
cases. Then

9x = yi> ^2 = ^2' ^^^^ ^s *° ^^y ^2 = 2^1-

If now we reduce the velocity of the motion in II
one-half, so that the velocities in I and II become
equal, (p^ will thereby be reduced one-fourth, that is
to say to ^1/2. Generalising, we get this rule : when
the velocity of the circular motion is the same^ the cen-
tripetal acceleration is inversely proportional to the
radius of the circle described.

4. The early investigators, owing to their following

i6o

THE SCIENCE OF MECHANICS,

Deduction the conceptions of the ancients, generally obtained their
era! law of propositions in the cumbersome form of proportions.

circular

motion. We shall pursue a different method. On a movable
object having the velocity v let a force act during the
element of time r which imparts to the object perpen-
dicularly to the direction of its motion the acceleration
<p. The new velocity-component thus becomes ^r,
and its composition with the first velocity produces a
new direction of the motion, making the angle a with
the original direction. From this results, by conceiving
the motion to take place in a circle of radius r, and on
account of the smallness of the angular element putting

Fir. 104. Fig. 105.

tan n' = nr, the following, as the complete expression
for the centripetal acceleration of a uniform motion in
a circle.

q}T

The para-
doxical
character
of this
problem.

7>2

= tan nr == fl' ^ -or<z;=: .
V r r

The idea of uniform motion in a circle conditioned
by a constant centripetal acceleration is a little para-
doxical. The paradox lies in the assumption of a con-
stant acceleration towards the centre without actual
approach thereto and without increase of velocity. This
is lessened when we reflect that without this centripetal
acceleration the body would be continually moving
away from the centre ; that the direction of the accel-

THE PRINCIPLES OF DYNAMICS. x6i

eration is constantly changing ; and that a change of
velocity (as will appear in the discussion of the prin-
ciple of vis viva) is connected with an approach of the
bodies that accelerate each other, which doe's not take
place here. The more complex case of elliptical cen-
tral motion is elucidative in this direction.

5. The expression for the centripetal or centrifugal a different

expression

acceleration, (p = r/^/r, can easily be put in a somewhat of the Uw.
different form. If T denote the periodic time of the
circular motion, the time occupied in describing the
circumference, then vT^=^^rn^ and consequently q) =
\rn^ jT^y in which form we shall employ the expres-
sion later on. If several bodies moving in circles have
the same periodic times, the respective centripetal ac-
celerations by which they are held in their paths, as is
apparent from the last expression, are proportional to
the radii.

6. We shall take it for granted that the reader is Some phe-
familiar with the phenomena that illustrate the con- which the

Ibw ex~

siderations here presented : as the rupture of strings of plains,
insufficient strength on which bodies are whirled about,
the flattening of soft rotating spheres, and so on. Huy-
gens was able, by the aid of his conception, to explain
at once whole series of phenomena. When a pendulum-
clock, for example, which had been taken from Paris
to Cayenne by Richer (1 671-1673), showed a retarda-
tion of its motion, Huygens deduced the apparent
diminution of the acceleration of gravity g thus estab-
lished, from the greater centrifugal acceleration of the
rotating earth at the equator ; an explanation that at
once rendered the observation intelligible.

An experiment instituted by Huygens may here be
noticed, on account of its historical interest. When
Newton brought out his theory of universal gravitation,

f6i THE SCIENCE OF MECHANICS.

An interest- Huygcns belonged to the great number of those who
ment of Were unable to reconcile themselves to the idea of action
uygens. ^^ ^ distance. He was of the opinion that gi^avitation
could be explained by a vortical medium. If we enclose
in a vessel filled with a liquid a number of lighter bod-
ies, say wooden balls in water, and set the vessel ro-
tating about its axis, the balls will at once rapidly move
towards the axis. If for instance (Fig. io6), we place
the glass cylinders RR containing the wooden balls KK
by means of a pivot Z on a rotatory apparatus, and ro-
tate the latter about its ver-
R . R^<\ tical axis, the balls will im-

mediately run up the cyl-
I I - inders in the direction away

M from the axis. But if the

a\ tubes be filled with water,

Fig. 106. each rotation will force the

balls floating at the extremities EE towards the axis.
The phenomenon is easily explicable by analogy with
the principle of Archimedes. The wooden balls receive
a centripetal impulsion, comparable to buoyancy,
which is equal and opposite to the centrifugal force
acting on the displaced liquid.
Oscillatory 7. Before we proceed to Huygens*s investigations

motion. . .

on the centre of oscillation, we shall present to the
reader a few considerations concerning pendulous and
oscillatory motion generally, which will make up in ob-
viousness for what they lack in rigor.

Many of the properties of pendulum motion were
known to Galileo. That he had formed the concep-
tion which we shall now give, or that at least he was
on the verge of so doing, may be inferred from many
scattered allusions to the subject in his Dialogues, The
bob of a simple pendulum of length / moves in a circle

THE PRINCIPLES OF D YNAMICS,

163

Fig. 107.

(Fig. 107) of radius /. If we give the pendulum a very Gauieps

small excursion, it will travel in its oscillations over ationofthe

very small arc which coincides approximately with the pendaium.

chord belonging to it. But this

chord is described by a falling

particle, moving oq it as on an

inclined plane (see Sect, i of this

Chapter, § 7), in the same time

as the vertical diameter BD =

2/. If the time of descent be

called /, we shall have 2/ =

\gt 2, that is / = 2 VTJg^ But

since the continued movement

from B up the line BC* occupies an equal interval of

time, we have to put for the time T of an oscillation

from Cto C\ T^= \V Ijg^ It will be seen that even from

so crude a conception as this the correct form of the

pendulum-laws is obtainable. The exact expression

for the time of very small oscillations is, as we know,

T=nvJlJ.

Again, the motion of a pendulum bob may be viewed pendaium
as a motion of descent on a succession of inclined wewed as a
planes. If the string of the pendulum makes the angle SoUn Ui-
a with the perpendicular, the pendulum bob receives pi^iCTes.
in the direction of the position of equilibrium the accel-
eration g, sin a. When a is small, ^. or is the expres-
sion of this acceleration ; in other words, the accelera-
tion is always proportional and oppositely directed to
the excursion. When the excursions are small the
curvature of the path may be neglected.

8. From these preliminaries, we may proceed to
the study of oscillatory motion in a simpler manner. A
body is free to move on a straight line OA (Fig. 108),
and constantly receives in the direction towards the

164

THE SCIENCE OF MECHANICS,

f^ ' y^

eye

A',

II y^

y^o'

B'A

Fig. 108.

A simpler point O an acceleration proportional to its distance from

view of OS- O, We will represent these accelerations by ordinates

motion. erected at the positions considered. Ordinates upwards

denote accelerations towards the left ; ordinates down-

wards represent accel-
erations towards the
right. The body, left
to itself at A, will
move towards O with
varied acceleration,
"B^A' P^ss through OioA^^
where OA^ = OA^
come back to O, and
so again continue its
motion. It is in the
The period first place easily demonstrable that the period of os-
tioninde- cillatiou (the time of the motion through AOA^) is in-
the ampii- dependent of the amplitude of the oscillation (the dis-
tance OA). To show this, let us imagine in I and
II the same oscillation performed, with single and
double amplitudes of oscillation. As the acceleration
varies from point to point, we must divide OA and
CA' :=20A into a very large equal number of ele-
ments. Each element A'B' of CyA' is then twice as
large as the corresponding element AB of OA. The
initial accelerations (p and (p' stand in the relation
(p' = 2^}. Accordingly, the elements AB and A'B* =
2 AB are described with their respective accelerations
(p and 2(p im, the same time r. The final velocities v
and v' in I and II, for the first element, will bev = (pr
and v' = 2(pXy that is z/' = 2 v. The accelerations and
the initial velocities at B and B' are therefore again as
1 : 2. Accordingly, the corresponding elements that
next succeed will be described in the same time. And

THE PRINCIPLES OF D YNAMICS.

165

of every succeeding pair of elements the same asser-
tion also holds true. Therefore, generalising, it will
be readily perceived that the period of oscillation is
independent of its amplitude or breadth.

Next, let us conceive two oscillatory motions, I and The time of

. . oscillation

II, that have equal excursions (Fig. log); but in II let inversely

, proportion-

a fourfold acceleration correspond to the same distance ai to the

sauare root
of the ac-
celeration.

^-r^

\^^o

II /

/O'

BA'

Fig. X09.

from O, We divide the amplitudes of
the oscillations AO and OA* = OA
into a very large equal number of
parts. These parts are then equal in
I and II. The initial accelerations at
A and A* are <p and 4 (p ; the ele-
ments of the distance described are
AB = A'B* = j; and the times are
respectively r and r*. We obtain, then,
r = \/2s/<p, T* = 1/2 j/4 (p = t/2.
The element A'B* is accordingly trav-
elled through in one-half the time
the element AB is. The final velocities v and v' at
B and B are found by the equations v ^= <pr and
z;' = 4 q}(j/2) =:2v. Since, therefore, the initial velo-
cities at B and B are to one another as i : 2, and the
accelerations are again as 1:4, the element of II suc-
ceeding the first will again be traversed in half the
time of the corresponding one in I. Generalising, we
get : For equal excursions the time of oscillation is in-
versely proportional to the square root of the accelera-
tions.

9. The considerations last presented may be put in
a very much abbreviated and very obvious form by a
method of conception first employed by Newton. New;
ton calls those material systems similar that have geo-
metrically similar configurations and whose homolo-

i66 THE SCIENCE OF MECHANICS,

The Diinci- gous masses bear to one another the same ratio. He
Uude. says further that systems of this kind execute similar
movements when the homologous points describe simi-
lar paths in proportional times. Conformably to the
geometrical terminology of the present day we should
not be permitted to call mechanical structures of this
kind (of five dimensions) similar unless their homolo-
gous linear dimensions as well as the times and the
masses bore to one another the same ratio. The struc-
tures might more appropriately be termed affined to
one another.

We shall retain, however, the name phoronomically
similar structures, and in the consideration that is to
follow leave entirely out of account the masses.
In two such similar motions, then, let

the homologous paths be s and as^

the homologous times be / and ftt\ whence

the homologous velo-

cities are z/ = - and yz; = — — ,

/ at

the homologous accel- ^ «

erations y, = ^_ and £^ = — — .

Thededac- Now all oscillations which a body performs under
laws of o^ the conditions above set forth with any two different
this^metho^ amplitudes I and a, will be readily recognised as sim-
ilar motions. Noting that the ratio of the homologous
accelerations in this case is f = at, we have a = a/ft^.
Wherefore the ratio of the homologous times, that is
to say of the times of oscillation, is )5 = zfc i. We ob-
tain thus the law, that the period of oscillation is inde-
pendent of the amplitude.

If in two oscillatory motions we put for the ratio
between the amplitudes i : a, and for the ratio between
the accelerations i : a/i, we shall obtain for this case

THE PRINCIPLES OF D YNAMICS.

167

£== ar;i= a/fi^y and therefore )5= i/zt ^/ /i ; where-
with the second law of oscillating motion is obtained.

Two uniform circular motions are always phoronom-
ically similar. Let the ratio of their radii be i : a and
the ratio of their velocities i : y. The ratio of their
accelerations is then £= flf//?*, and since y=a/fi,
3\so€=i y^/a ; whence the theorems relative to cen-
tripetal acceleration are obtained.

It is a pity that investigations of this kind respect-
ing mechanical and phoronomical affinity are not more
extensively cultivated, since they promise the most
beautiful and most elucidative extensions of insight
imaginable.

10. Between uniform motion in a circle and oscil-
latory motion of the kind just discussed an important
relation exists which we shall now consider. We as-
sume a system of rectangular co-
ordinates, having its origin at the
centre, Oy of the circle of Fig. no,
about the circumference of which
we conceive a body to move uni-
formly. The centripetal accelera-
tion q> which conditions this mo-
tion, we resolve in the directions
of X and Y\ and observe that the -Y-components of the
motion are affected only by the A'-components of the
acceleration. We may regard both the motions and
both the accelerations as independent of each other.

Now, the two components of the motion . are os-
cillatory motions to and fro about O. To the excur-
sion X the acceleration-component q) (jx/r) or (^<pfr) x
in the direction O, corresponds. The acceleration is
proportionaly therefore, to the excursion. And accord-
ingly the motion is of the kind just investigated. The

The con-
nection be-
tween oscil-
latory mo-
tion of this
kind and
nniform
motion in a
circle.

The iden-
tity of the
two.

i68 THE SCIENCE OF MECHANICS.

time 7* of a complete to and fro movement is also the
periodic time of the circular motion. With respect to
the latter, however, we know that ^= ^rn^ /T^, or,
what is the same, that T^=iin \^r/(p. Now <p/r is
the acceleration for j:= i, the acceleration that corre-
sponds to unit of excursion, which we shall briefly
designate by /. For the oscillatory motion we may
put, therefore, T=z 2n\/i//, For a single movement
to, or a single movement fro, — the common method of
reckoning the time of oscillation, «— we get, then, Tz=z
nVTif,
The appu- 1 1 • Now this result is directly applicable to pen-
the lasfre- dulum vibratious of very small excursions, where, ne-
duium^" glecting the curvature of the path, it is possible to ad-
ra ions, j^^^.^ ^^ ^^ conception developed. For the angle of

elongation a we obtain as the distance of the pendulum
bob from the position of equilibrium, la ; and as the
corresponding acceleration, ga\ whence

/=^-^ = -5andr=;rJ-^.
'' la I \ g

This formula tells us, that the time of vibration is
directly proportional to the square root of the length
of the pendulum, and inversely proportional to the
square root of the acceleration of gravity. A pendulum
that is four times as long as the seconds pendulum,
therefore, will perform its oscillation in two seconds.
A seconds pendulum removed a distance equal to the
earth's radius from the surface of the earth, and sub-
jected therefore to the acceleration gl\^ will likewise
perform its oscillation in two seconds.

12. The dependence of the time of oscillation on
the length of the pendulum is very easily verifiable by
experiment. If (Fig. iii) the pendulums a, ^, r,

THE PRIKCIPLES OF DYNAMICS. 169

. which to maintain the plane of oscillation invariable Sipninen-
are suspended by double threads, have the lengths i, tiaaoftbe
4, 9, then a will execute two oscillations to one oscil- penduiam.
lation of b, and three to one of c.

The verification of the dependence of the time of
oscillation on the acceleration of gravity g is some-
what more difficult ; since the latter cannot be arbi-
trarily altered. But the demonstration can be effected
by allowing one component only of g to act on the
pendulum. If we imagine the axis of oscillation of

I70 THE SCIENCE OF MECHANICS.

a- the pendulum A A fixed in the vertically placed plaae

I of the paper, EE will be the intersection of the plane

A^p of oscillation with the plane of the paper

y ~~~^ and likewise the position of equilibrium

L of the pendulum. The axis makes with

/ the horizontal plane, and the plane of os-

/ cillation makes with the vertical plane, the

^ angle /S; wherefore the acceleration ^.cos^

Fig. III. js {hg acceleration which acts in this plane.

!f the pendulum receive in the plane of its oscillation

the small elongation a, the corresponding acceleration

■will be {g cos p) a ; whence the time of oscillation ii

THE PRINCIPLES OF D YNAMICS,

171

We see from this result, that as fi is increased the
acceleration g cos ft diminishes, and consequently the
time of oscillation increases. The experiment may be
easily made with the apparatus represented in Fig. 113.
The frame RR is free to turn about a hinge at C; it can
be inclined and placed on its side. The angle of in-
clination is fixed by a graduated arc G held by a set-
screw. Every increase of ft increases the time of oscil-
lation. If the plane of oscillation be made horizontal,
in which position R rests on the foot F^ the time of
oscillation becomes infinitely great. The pendulum
in this case no longer returns to any definite position
but describes several complete revolutions in the same
direction until its entire velocity has been destroyed
by friction.

13. If the movement of the pendulum do not take The conical
place in a plane, but be performed in space, the thread
of the pendulum will describe the surface
of a cone. The motion of the conical pen-
dulum was also investigated byHuygens.
We shall examine a simple case of this
motion. We imagine (Fig. 114) a pen-
dulum of length / removed from the ver- ^
tical by the angle or, a velocity «; imparted
to the bob of the pendulum at right
angles to the plane of elongation, and the pendulum re-
leased. The bob of the pendulum will move in a hori-
zontal circle if the centrifugal acceleration cp developed
exactly equilibrates the acceleration of gravity g\ that
is, if the resultant acceleration falls in the direction of
the pendulum thread. But in that case <p/g=^ tan or.
If 7^ stands for the time taken to describe one revolu-
tion, the periodic time, then ^ = 4r;r2/7^ or 7^ =
2 7t Vr/(p, Introducing, now, in the place of r/(p the

Fig. 114.

172 THE SCIENCE OF MECHANICS,

value / sin ajg tan a z=ii cos a/gy we get for the periodic
time of the pendulum, T:=2 7C V i cos a/g. For the ve-
locity V of the revolution we find v = l/r<^, and since
(p = ^tan a it follows that v = V^gl sin a Ian a. For
very small elongations of the conical pendulum we may
put Tz=in V llg, which coincides with the regular
formula for the pendulum, when we reflect that a single
revolution of the conical pendulum corresponds to two
vibrations of the common pendulum.
The deter- 1 4. Huygens was the first to undertake the exact
the accei- determination of the acceleration of gravity g by means
gravity by of pcudulum observations. From the formula 7*=
lumf*" " n V Tig for a simple pendulum with small bob we ob-
tain directly ^=: n^ IjT^, For latitude 45° we obtain
as the value of g^ in metres and seconds, 9 . 806. For
provisional mental calculations it is sufficient to re-
member that the acceleration of gravity amounts in
round numbers to 10 metres a second.
A remark 1 5. Every thinking beginner puts to himself the

pia express- question how it is that the duration of an oscillation,

iDflf thfi 1a.w

■ that is a time^ can be found by dividing a number that
is the measure of a length by a number that is the
measure of an acceleration and extracting the square
root of the quotient. But the fact is here to be borne in
mind that^=2j//2, that is a length divided by the
square of a time. In reality therefore the formula we
have is 7^= n '\/{l/2s)t^. And since I /2 s is the ratio
of two lengths, and therefore a number, what we have
under the radical sign is consequently the square of a
time. It stands to reason that we shall find T'in sec-
onds only when, in determining g, we also take the sec-
ond as unit of time.

In the formula gz=zn^ IjT^ we see directly that g is

THE PRINCIPLES OF DYNAMICS. 173

a length divided by the square of a time, according to

the nature of an acceleration.

16. The most important achievement of Huygens The prob-
lem of the
is his solution of the problem to determine the centre centre of

oscillation.

of oscillation. So long as we have to deal with the dy-
namics of a single body, the Galilean principles amply
suffice. But in the problem just mentioned we have to
determine the motion of several bodies that mutually
influence each other. This cannot be done without
resorting to a new principle. Such a one Huygens
actually discovered.

We know that long pendulums perform their oscil- statement

.of the prob-

lations more slowly than short ones. Let us imagine a lem.
heavy body, free to rotate about an axis, the centre of
gravity of which lies outside of the axis ; such
a body will represent a compound pendulum.
Every material particle of a pendulum of this
kind would, if it were situated alone at the
same distance from the axis, have its own pe-
riod of oscillation. But owing to the connec- Fig. ns.
tions of the parts the whole body can vibrate with only
a single, determinate period of oscillation. If we pic-
ture to ourselves several pendulums of unequal lengths,
the shorter ones will swing quicker, the longer ones
slower. If all be joined together so as to form a single
pendulum, it is to be presumed that the longer ones
will be accelerated, the shorter ones retarded, and that
a sort of mean time of oscillation will result. There
must exist therefore a simple pendulum, intermediate
in length between the shortest and the longest, that
has the same time of oscillation as the compound pen-
dulum. If we lay off the length of this pendulum on
the compound pendulum, we shall find a point that pre-
serves the same period of oscillation in its connection

174 ^^^^ SCIENCE OF MECHANICS.

with the other points as it would have if detached and
left to itself. This point is the centre of oscillation.
Mersenne was the first to propound the problem of
determining the centre of oscillation. The solution of
Descartes, who attempted it, was, however, precipi-
tate and insufficient.
Huy^ens's 1 7. Huvgeus was the first who gave a general solu-

SOlutlOn. -r^ • 1 ▼▼ t It t • •

tion. Besides Huygens nearly ail the great inquirers
of that time employed themselves on the problem, and
we may say that the most important principles of mod-
ern mechanics were developed in connection with it.

The new idea from which Huygens set out, and
which is more important by far than the whole prob-
lem, is this. In whatsoever manner the material par-
ticles of a pendulum may by mutual interaction modify
each other's motions, in every case the velocities ac-
quired in the descent of the pendulum can be such only
that by virtue of them the centre of gravity of the par-
ticles, whether still in connection or with their connec-
tions dissolved, is able to rise just as high as the point
The new from which xtfelL Huygens found himself compelled,
whfch^Huy- ^Y ^^ doubts of his contemporaries as to the correct-
Suced?''**^ ness of this principle, to remark, that the only assump-
tion implied in the principle is, that heavy bodies of
themselves do not move upwards. If it were possible
for the centre of gravity of a connected system of falling
material particles to rise higher after the dissolution
of its connections than the point from which it had
fallen, then by repeating the process heavy bodies
could, by virtue of their own weights, be made to rise
to any height we wished. If after the dissolution of
the connections the centre of gravity should rise to a
height less than that from which it had fallen, we
should only have to reverse the motion to produce the

THE PRINCIPLES OF DYNAMICS, 175

same result. What Huygens asserted, therefore, no
one had ever really doubted ; on the contrary, every
one had instinctively perceived it. Huygens, however,
gave this instinctive perception an abstract^ conceptual
form. He does not omit, moreover, to point out, on the
ground of this view, the fruitlessness of endeavors to
establish a perpetual motion. The principle just devel-
oped will be recognised as a generalisation of one of Ga-
lileo's ideas,

18. Let us now see what the principle accomplishes Huygens's

principle

in the determination of the centre of oscillation. Let applied-
OA (Fig. 116), for simplicity's sake,
be a linear pendulum, made up of a
large number of masses indicated in
the diagram by points. Set free at
OA^ it will swing through B to 0A\
where AB = BA', Its centre of
gravity S will ascend just as high
on the second side as it fell on the
first. From this, so far, nothing would follow. But
also, if we should suddenly, at the position OBy re-
lease the individual masses from their connections, the
masses could, by virtue of the velocities impressed on
them by their connections, only attain the same height
with respect to centre of gravity. If we arrest the free
outward-swinging masses at the greatest heights they
severally attain, the shorter pendulums will be found
below the line OA', the longer ones will have passed
beyond it, but the centre of gravity of the system will
be found on OA' in its former position.

Now let us note that the enforced velocities are
proportional to the distances from the axis ; therefore,
one being given, all are determined, and the height of
ascent of the centre of gravity given. Conversely,

1/6

THE SCIENCE OF MECHANICS,

therefore, the velocity of any material particle also is
determined by the known height of the centre of grav-
ity. But if we know in a pendulum the velocity cor-
responding to a given distance of descent, we know its
whole motion.
Thede- 19. Premising these remarks, we proceed to the

tailed reso- .

lution of the problem itself. On a compound linear pendulum (Fig.

problein.

117) we cut off, measuring from the axis, the
portion =1. If the pendulum move from its
position of greatest excursion to the position
of equilibrium, the point at the distance = i
from the axis will fall through the height k.
The masses m^ tn\ m" , , , at the distances
r, r', r" . . . will fall in this case the dis-
tances rky r' ky r'* k , . . , and the distance of
the descent of the centre of gravity will be :

Fig. X17.

mrk 4- m'r'k -f m**r"k -f . . . _,^mr
m -\- m' -\- m" ^ . . , . ^m

Let the point at the distance i from the axis ac-
quire, on passing through the position of equilibrium,
the velocity, as yet unascertained, v. The height of
its ascent, after the dissolution of its connections, will
be v^/2g. The corresponding heights of ascent of
the other material particles will then be {fvY/2g,
(r' v) 2/2^, (r" vy /2g . . . . The height of ascent of the
centre of gravity of the liberated masses will be

^g ^g ^g

7^^ 2mr^
— 2g 2 m

fft -\- fn -\- tn -f- . . .

By Huygens's fundamental principle, then^

v^ 2mr^
2g '2f^

2mr

(«).

THE PRINCIPLES OF D YNAMICS. 177

From this a relation is deducible between the distance of
descent k and the velocity v. Since, however, all pen-
dulum motions of the same excursion are phoronomi-
cally similar, the motion here under consideration is,
in this result, completely determined.

To find the length of the simple pendulum that has The length

of the siixi'

the same period of oscillation as the compound pen- pie iaoch-

ronoQS

dulum considered, be it noted that the same relation penduiam.
must obtain between the distance of its descent and its
velocity, as in the case of its unimpeded fall. If y is
the length of this pendulum, ky is the distance of its
descent, and vy its velocity ; wherefore

y-tg-i^ w-

Multiplying equation {a) by equation (^) we obtain

Js mr

Employing the principle of phoronomic similitude, solution of
we may also proceed in this way. From {a) we get iem^7the

principle of

"^Plf unulitnde.

A simple pendulum of length i, under corresponding
circumstances, has the velocity

Calling the time of oscillation of the compound pendu-
lum y, that of the simple pendulum of length i T^ =
nV'i/gj we obtain, adhering to the supposition of
equal excursions,

--=: » ; wherefore 7^= tt a —v. -•
T^ V \ g2 m r

v=Vigk^

178 THE SCIENCE OF MECHANICS.

Hnygens's 20. We scc without difficulty in the Huygenian

principle ..i, •• r > i j--*

Identical principle the recognition of work as the condition de-
principle of terminative of velocity^ or, more exactly, the condition
determinative of the so-called vis viva. By the vis
viva or living force of a system of masses m^ m„
m,, . . . ., affected with the velocities v, v„ v,, , . , ., we
understand the sum *

The fundamental principle of Huygens is identical with
the principle of vis viva. The additions of later in-
quirers were made not so much to the idea as to the
form of its expression.

If we picture to ourselves generally any system of
weights/,/,,/,, . . . ., which fall connected or uncon-
nected through the heights h, ^„ ^,, . . . ., and attain
thereby the velocities v, v„ v,, , . . ., then, by the Huy-
genian conception, a relation of equality exists between
the distance of descent and the distance of ascent of the
centre of gravity of the system, and, consequently, the
equation holds

7/2

/+/ + />"

+ •

2. -^ .

or 2/f/i ■

If we have reached the concept of "mass," which
Huygens did not yet possess in his investigations, we
may substitute iot pjg the mass m and thus obtain the
form '2ph= ^2 mv^, which is very easily generalised
for non-constant forces.

* This is not the usual definition of English writers, who follow the older
authorities in making the vis viva twice this quantity. — Trans.

THE PRINCIPLES OF DYNAMICS. 179

21. With the aid of the principle of living forces General

. . . method of

we can determine the duration of the infinitely small determin-
ing the pe-

oscillations of any pendulum whatso- / s^ nodofpen-

tf / ^""^^ - dnlnm os-

ever. We let fall from the centre of ( ^ \ ciUationa.

gravity s (Fig. 1 18) a perpendicular on
the axis ; the length of the perpendic-
ular is, say, a. We lay off on this,
measuring from the axis, the length
= I. Let the distance of descent of
the point in question to the position of ^**' "®-

equilibrium be ky and v the velocity acquired. Since
the work done in the descent is determined by the
motion of the centre of gravity, we have

work done in descent = vis viva :

akgM=^~gr- 2mr^,

M here we call the total mass of the pendulum and
anticipate the expression vis viva. By an inference
similar to that in the preceding case, we obtain 7'=
n V^mr^/agM'

22. We see^. that the duration of infinitely small The two

"* determma-

oscillations of any pendulum is determined by two fac- tive factors,
tors — by the value of the expression 2mr^, which
Euler called the moment 0/ inertia and which Huygens
had employed without any particular designation, and
by the value of agM, The latter expression, which we
shall briefly term the statical moment , is the product
aP oi the weight of the pendulum into the distance of
its centre of gravity from the axis. If these two values
be given, the length of the simple pendulum of the
same period of oscillation (the isochronous pendulum)
and the position qf the centre of oscillation are deter-
mined.

z8o

THE SCIENCE OF MECHANICS,

Fig. 1 19.

r, r

HajrgeoB's Fof the determination of the lengths of the pendu-

methodBof lums referred to, Huygens, in the lack of the analytical

methods later discovered, employed a very ingenious

geometrical procedure, which
we shall illustrate by one or
two examples. Let the prob-
lem be to determine the time
of oscillation of a homogene-
ous, material, and heavy rec-
tangle ABCD^ which swings
on the axis AB (Fig. 119).
Dividing the rectangle into
minute elements of zr^^ifyf,,
/„.... having the distances
from the axis, the expression for the
length of the isochronous simple pendulum, or the dis-
tance of the centre of oscillation from the axis, is given
by the equation

A+/, ^+/,^, +

Let us erect on A BCD at C and D the perpendiculars
CE'=^ DF^=^AC=^ BD and picture to ourselves a
homogeneous wedge ABCDEF, Now find the distance
of the centre of gravity of this wedge from the plane
through AB parallel to CDEF. We have to consider,
in so doing, the tiny columns /r,/, r^yf^^r,^ . . . . and
their distances r^r^^ r^^ . , . . from the plane referred
to. Thus proceeding, we obtain for the required dis-
tance of the centre of gravity the expression

/r. r+/,r, . r, +/,,r, , . r,, + . , , .

that is, the same expression as before. The centre of
oscillation of the rectangle and the centre of gravity of

THE PRINCIPLES OF D YNAMICS.

i8i

the wedge are consequently at the same distance from
the axis, \AC.

Following out this idea, we readily perceive the Analogous
correctness of the following assertions. For a homo- tions or the
geneous rectangle of height h swinging about one of methods,
its sides, the distance of the centre of gravity from the
axis is h/2f the distance of the centre of oscillation | h»
For a homogeneous triangle of height h, the axis of
which passes through the vertex parallel to the base,
the distance of the centre of gravity from the axis is
\hf the distance of the centre of oscillation ^h. Call-
ing the moments of inertia of the rectangle and of the
triangle J^, J^' ^^^ their respective masses Af^^ J/^,
we get

2^»
Consequently ^ j = — -„-^

. ^, = "2 '-

By this pretty geometrical conception many prob-
lems can be solved that are to-day treated — more con-
veniently it is true — by routine forms.

Fig. 120. Fig. 121.

23. We shall now discuss a proposition relating to
moments of inertia, that Huygens made use of in a
somewhat different form. Let O (Fig. 121) be the
centre of gravity of any given body. Make this the

i82 THE SCIENCE OF MECHANICS.

The reia- origin of a system of rectangular coordinates, and sup-

tion of ino~

ments of in- pose the moment of inertia with reference to the Z-axis

ertia re-
ferred to determined. If m is the element of mass and r its dis-

parallel

axes. tance from the Z-axis, then this moment of inertia is

^ = 2mr^, We now displace the axis of rotation
parallel to itself to (7, the distance a in the Jf-direction.
The distance r is transformed, by this displacement,
into the new distance p, and the new moment of
inertia is '

0 = 2mp^ = 2m [(^ — ay +y^'] = 2m (jc* + y^) —
2a2mX'\'a^2m, or, since 2 m (^x^ -\-y^) ^ 2 m r^ =^,
calling the total mass M= 2m, and remembering the
property of the centre of gravity 2mx = 0,

From the moment of inertia for one axis through the
centre of gravity, therefore, that for any other axis
parallel to the first is easily derivable.
Anappii- 24. An additional observation presents itself here.

cation of ^ ■"^■^

thispropo- The distance of the centre of oscillation is given by

sition. ^ -^

the equation lz=,A-\- a^M/aMy where z^. My and a
have their previous significance. The quantitiess^ and
M are invariable for any one given body. So long
therefore as a retains the same value, / will also remain
invariable. For all parallel axes situated at the same
distance from the centre of gravity, the same body as
pendulum has the same period of oscillation. If we
put ^/M=^ Xy then

/^ \- a.

Now since / denotes the distance of the centre of
oscillation, and a the distance of the centre of gravity
from the axis, therefore the centre of oscillation is
always farther away from the axis than the centre of

THE PRINCIPLES OF D YNAMICS. 183

gravity by the distance x/a. Therefore x/a is the dis-
tance of the centre of oscillation from the centre of
gravity. If through the centre of oscillation we place
a second axis parallel to the original axis, a passes
thereby into x/a, and we obtain the new pendulum
length

X a a

The time of oscillation remains the same therefore
for the second parallel axis through the centre of oscil-
lation, and consequently the same also for every par-
allel axis that is at the same distance x/a from the
centre of gravity as the centre of oscillation.

The totality of all parallel axes corresponding to
the same period of oscillation and having the distances a
and x/a from the centre of gravity, is consequently re-
alised in two coaxial cylinders. Each generating line
is interchangeable as axis with every other generating
line without affecting the period of oscillation.

25. To obtain a clear view of the relations subsist- The axial
ing between the two axial cylinders, as we shall briefly
call them, let us institute the following considerations^
We put A z=k^M, and then

a
If we seek the a that corresponds to a given /, and
therefore to a given time of oscillation, we obtain

= i^^1-''-

Generally therefore to one value of / there correspond
two values of a. Only where V l^ /\ — ^2 __ q^ that is
in cases in which 1= ik, do both values coincide in
a = k.

184

THE SCIENCE OF MECHANICS.

If we designate the two values of a that correspond
to every /, by a and /?, then

/= = — - - ,or

flt p

The deter- If, therefore, in any pendulous body we know two par-

mmation of ,

the preced- allel axes that have the same time of oscillation and

ing factors , z» r

by a geo- different distances a and p from the centre of gravity,

metrical . . ' , , • /

method, as IS the case for instance where we are able to give the
centre of oscillation for any point of suspension, we
can construct k. We lay off (Fig. 122) a and P con-

secutively on a straight line, describe a semicircle on
a -\- p ^s diameter, and erect a perpendicular at the
point of junction of the two divisions a and p. On this
perpendicular the semicircle cuts oH k. If on the other
hand we know k, then for every value of a, say A, a
value fx is obtainable that will give the same period
of oscillation as X, We construct (Fig. 123) with \
and k as sides a right angle, join their extremities by a
straight line on which we erect at the extremity of /& a
perpendicular which cuts off on X produced the por-
tion jU.

Now let us imagine any body whatsoever (Fig. 124)
with the centre of gravity O, We place it in the plane

THE PRINCIPLES OF DYNAMICS. 185

of the drawing, and make it swine about all possible An iiiaitn-
parallel axes at right angles to the plane of the paper, idn.
All the axes that pass through the circle tx are, we
find, with respect to period of oscillation, interchange-
able with each other and also with those that pass
through the circle ji. If instead of a we take a smaller
circle X, then in the place of /? we shall get a larger

Fig. Ill,

circle. }i. Continuing in this manner, both circles ul-
timately meet in one with the radius k.

26. We have dwelt at such length on the foregoing Bec»pi«oi»-
matters for good reasons. In the first place, they have
served our purpose of displaying in a clear light the
splendid results of the investigations of Huygens. For
all that we have given is virtually contained, though
in somewhat different form, in the writings of Huygens,

i86 THE SCIENCE OF MECHANICS.

or is 'at least so approximately presented in them that
it can be supplied without the slightest difficulty. Only
a very small portion of it has found its way into our
modern elementary text-books. One of the proposi-
tions that has thus been incorporated in our elemen-
tary treatises is that referring to the convertibility of
the point of suspension and the centre of oscillation.
The usual presentation, however, is not exhaustive.
Captain Kater, as we know, employed this principle
for determining the exact length of the seconds pen-
dulum.
Function of The points raised in the preceding paragraphs have

the nioniont

of inertia, also rendered us the service of supplying enlighten-
ment as to the nature of the conception '^ moment of
inertia." This notion affords us no insight, in point
of principle, that we could not have obtained without
it. But since we save by its aid the individual con-
sideration of the particles that make up a system, or
dispose of them once for all, we arrive by a shorter
and easier way at our goal. This idea, therefore, has
a high import in the economy of mechanics. Poinsot,
after Euler and Segner had attempted a similar object
with less success, further developed the ideas that be-
long to this subject, and by his ellipsoid of inertia and
central ellipsoid introduced further simplifications.
The lesser 27. The investigations of Huygens concerning the
tfoM of* geometrical and mechanical properties of cycloids are
of less importance. The cycloidal pendulum, a contriv-
ance in which Huygens realised, not an approximate,
but an exact independence of the time and amplitude
of oscillation, has been dropt from the practice of mod-
ern horology as unnecessary. We shall not, therefore,
enter into these investigations here, however much of
the geometrically beautiful they may present.

Huygens.

THE PRINCIPLES OF D YNAMICS. 187

Great as the merits of Huygens are with respect to Hnygens's
the most different physical theories, the art of horology, achieve-
practical dioptrics, and mechanics in particular, his
chief performance, the one that demanded the greatest
intellectual courage, and that was also accompanied
with the greatest results, remains his enunciation of the
principle by which he solved the problem of the centre
of oscillation. This very principle, however, was the
only one he enunciated that was not adequately appre-
ciated by his contemporaries ; nor was it for a long
period thereafter. We hope to have placed this prin-
ciple here in its right light as identical with the prin-
ciple of vis viva,

III.

THE ACHIEVEMENTS OF NEWTON.

1. The merits of Newton with respect to our sub- Newton's
ject were twofold. First, he greatly extended the range

of mechanical physics by his discovery of universal
gravitation. Second, he completed the formal enunciation
of the mechanical principles now generally accepted. Since
his time no essentially new principle has been stated.
All that has been accomplished in mechanics since his
day, has been a deductive, formal, and mathematical
development of mechanics on the basis of Newton's
laws.

2. Let us first cast a glance at Newton's achieve- His ^eat
ment in the domain of physics, Kepler had deduced discovery,
from the observations of Tycho Brahe and his own,

three empirical laws for the motion of the planets
about the sun, which Newton by his new view rendered
intelligible. The laws of Kepler are as follows :
i) The planets move about the sun in ellipses, in
one focus of which the sun is situated.

i88 THE SCIENCE OF MECHANICS.

Kepler's 2) The radius vector joining each planet with the

law.. Their ^ , .. , • w

part inthe sun describes equal areas m equal times.

3) The cubes of the mean distances of the planets
from the sun are proportional to the squares of
their times of revolution.
He ivho clearly understands the doctrine of Galileo
and HuygenSy must see that a curvilinear motion im-
plies deflective acceleration. Hence, to explain the phe-
nomena of planetary motion, an acceleration must be
supposed constantly directed towards the concave side
of the planetary orbits.
Central ac- Now Kepler's second law, the law of areas, is ex-

celeration , . , , , . - ,

explains plained at once by the assumption of a constant plane-

Kepler's ^ , • j , , , •

aecond law. tary acceleration towards the sun ; or rather, this ac-
celeration is another form of expression for the same

fact. If a radius vector describes
in an element of time the area
ABS (Fig. 125), then in the next
equal element of time, assuming
no acceleration, the area BCS
will be described, where BC =
AB and lies in the prolongation
Fig. us. of AB, But if the central accel-

eration during the first element of time produces a
velocity by virtue of which the distance BD will be
traversed in the same interval, the next-succeeding
area swept out is not BCS^ but BES, where CE is par-
allel and equal to BD, But it is evident that BES =
BCS = ABS, Consequently, the law of the areas con-
stitutes, in another aspect, a central acceleration.

Having thus ascertained the fact of a central accel-
eration, the third law leads us to the discovery of its
character. Since the planets move in ellipses slightly
different from circles, we may assume, for the sake of

THE PRINCIPLES OF DYNAMICS. 189

simplicity, that their orbits actually are circles. If ^1, The formal
R^y R^ are the radii and T^^ T^^ T^ the respective of this ac-
times of revolution of the planets, Kepler's third law deducibie

, . - ,, from Kep-

may be written as follows : ler's third

law.

R ^ R ^ R ^

^= 7^ = -^=.. =a constant.

But we know that the expression for the central accel-
eration of motion in a circle is (p= \R7t^ /T^, or
7^ = 4 TT^ R/(p. Substituting this value we get

(p^R^^ ^= (p^R^^ = <^3 ^3^ = constant ; or

<p = constant /R"^ ;

that is to say, on the assumption of a central accelera-
tion inversely proportional to the square of the distance,
we get, from the known laws of central motion, Kep-
ler's third law ; and vice versa.

Moreover, though the demonstration is not easily
put in an elementary form, when the idea of a central
acceleration inversely proportional to the square of the
distance has been reached, the demonstration that this
acceleration is another expression for the motion in
conic sections, of which the planetary motion in ellipses
is a particular case, is a mere affair of mathematical
analysis.

3. But in addition to the intellectual performance The qaes-
just discussed, the way to which was fully prepared by physical

character of

Kepler, Galileo, and Huygens, still another achieve- this accei-
ment of Newton remains to be estimated which in no
respect should be underrated. This is an achievement
of the imagination. We have, indeed, no hesitation
in saying that this last is the most important of all.
Of what nature is the acceleration that conditions the
curvilinear motion of the planets about the sun, and
of the satellites about the planets ?

I90 THE SCIENCE OF MECHANICS.

The steps Newton oerceived, with great audacity of thought,

which orig- .

inaiiy led and first in the instance of the moon, that this accel-
the idea of eration differed in no substantial respect from the ac-

universal ..-.-.,. , ,

gravitation, celeration of gravity so familiar to us. It was prob-
ably the principle of continuity, which accomplished
so much in Galileo's case, that led him to his dis-
covery. He was wont — and this habit appears to be
common to all truly great investigators — to adhere as
closely as possible, even in cases presenting altered
conditions, to a conception once formed, to preserve
the same uniformity in his conceptions that nature
teaches us to see in her processes. That which is a
property of nature at any one time and in any one
place, constantly and everywhere recurs, though it
may not be with the same prominence. If the attrac-
tion of gravity is observed to prevail, not only on the
surface of the earth, but also on high mountains and in
deep mines, the physical inquirer, accustomed to con-
tinuity in his beliefs, conceives this attraction as also
operative at greater heights and depths than those ac-
cessible to us. He asks himself. Where lies the limit
of this action of terrestrial gravity ? Should its action
not extend to the moon ? With this question the great
flight of fancy was taken, of which, with Newton's in-
tellectual genius, the great scientific achievement was
but a necessary consequence.
The appii- Newtou discovered first in the case of the moon that
this idea to the Same acceleration that controls the descent of a
of the moon, stone also prevented this heavenly body from moving
away in a rectilinear path from the earth, and that, on
the other hand, its tangential velocity prevented it from
falling towards the earth. The motion of the moon
thus suddenly appeared to him in an entirely new light,
but withal under quite familiar points of view. The

THE PRINCIPLES OF D YNAMICS, igi

new conception was attractive in that it embraced ob-
jects that previously were very remote, and it was con-
vincing in that it involved the most familiar elements.
This explains its prompt application in other fields and
the sweeping character of its results.

Newton not only solved by his new conception the its univer-
sal applica-

thousand years' puzzle of the planetary system, buttiontoaii

matter.

also furnished by it the key to the explanation of a
number of other important phenomena. In the same
way that the acceleration due to terrestrial gravity ex-
tends to the moon and to all other parts of space, so do
the accelerations that are due to the other heavenly
bodies, to which we must, by the principle of contin-
uity, ascribe the same properties, extend to all parts
of space, including also the earth. But if gravitation is
not peculiar to the earth, its seat is not exclusively in the
centre of the earth. Every portion of the earth, how-
ever small, shares it. Every part of the earth attracts,
or determines an acceleration of, every other part.
Thus an amplitude and freedom of physical view were
reached of which men had no conception previously to
Newton's time. •

A long series of propositions respecting the action The sweep-
of spheres on other bodies situated beyond, upon, or ter of its re-
within the spheres ; inquiries as to the shape of the
earth, especially concerning its flattening by rotation,
sprang, as it were, spontaneously from this view. The
riddle of the tides, the connection of which with the
moon had long before been guessed, was suddenly ex-
plained as due to the acceleration of the mobile masses
of terrestrial water by the moon.

4. The reaction of the new ideas on mechanics was
a result which speedily followed. The greatly varying
accelerations which by the new view the same body be-

192 THE SCIENCE OF MECHANICS.

The effect came affected with according to its position in space,
ideas on suggested at once the idea of variable weight, yet also
' pointed to one characteristic property of bodies which
was constant. The notions of mass and weight were
thus first clearly distinguished. The recognised vari-
ability of acceleration led Newton to determine by spe-
cial experiments the fact that the acceleration of gravity
is independent of the chemical constitution of bodies ;
whereby new positions of vantage were gained for the
elucidation of the relation of mass and weight, as will
presently be shown more in detail. Finally, the uni-
versal applicability of Galileo's idea of force was more
palpably impressed on the mind by Newton's perform-
ances than it ever had been before. People could • no
longer believe that this idea was alone applicable to the
phenomenon of falling bodies and the processes most
immediately connected therewith. The generalisation
was effected as of itself, and without attracting partic-
ular attention. ^
Newton's 5. Let US now discuss, more in detail, the achieve-
ments in ments of Newton as they bear upon the principles of
of mechan- mechanics. In so doing/ we shall first devote ourselves
exclusively to Newton's ideas, seek to- bring them for-
cibly home to the reader's mind, and restrict our criti-
cisms wholly to preparatory remarks, reserving the
criticism of details for a subsequent section. On pe-
rusing Newton's work {J^hilosophice Naturalis Principia
Mathematica, London, 1687), the following things
strike us at once as the chief advances beyond Galileo
and Huygens :

i) The generalisation of the idea of force.

2) The introduction of the concept of mass.

3) The distinct and general formulation of the prin-
ciple of the parallelogram of forces.

THE PRINCIPLES OF D YNAMICS. 193

4) The statement of the law of action and reaction.
6. With respect to the first point little is to beHUattitade
added to what has already been said. Newton con- to the idea

... , . . - . offeree.

ceives all circumstances determinative of motion,
whether terrestrial gravity or attractions of planets, or
the action of magnets, and so forth, as circumstances
determinative of acceleration. He expressly remarks
on this point that by the words attraction and the like
he does not mean to put forward any theory concern-
ing the cause or character of the mutual action referred
to, but simply wishes to express (as modern writers
say, in a differential form) what is otllerwise expressed
(that is, in an integrated form) in the description of the
motion. Newton's reiterated and emphatic protesta-
tions that he is not concerned with hypotheses as to the
causes of phenomena, but has simply to do with the
investigation and transformed statement of actual facts^
— a direction of thought that is distinctly and tersely
uttered in his words ** hypotheses non fingo," *'I do
not frame hypotheses," — stamps him as a philosopher
of the highest rank. He is not desirous to astound and The rcru-
startle, or to impress the imagination by the originality phandi.
of his ideas : his aim is to know Nature, *

* This is conspicacusly shown in the rules that Newton formed for the
conduct of natural inquiry (the Regula Pkilotopkandi) :

" Rule I. No more causes of natural things are to be admitted than such
as truly exist and are sufficient to explain the phenomena of these things.

" Rule II. Therefore, to natural effects of the same kind we must, as far
as possible, assign the same causes; e. g., to respiration in man and animals ;
to the descent of stones in Europe and in America ; to the light of our kitchen
fire and of the sun ; to the reflection of light on the earth and on the planets.

*' Rnle III. Those qualities of bodies that can be neither increased nor
diminished, and which are found to belong to all bodies within the reach of
our experiments, are to be regarded as the universal qualities of all bodies.
[Here follows the enumeration of the properties of bodies which has been in-
corporated in all text-books.] *

** If it universally appear, by experiments and astronomical observations,
that all bodies in the vicinity of the earth are heavy with respect to the earth,
and this in proportion to the quantity of matter which they severally contain ;

194

THE SCIENCE OF MECHANICS,

The New-
tonian con-
cept of
mass.

The expe-
riences
which point
to the exist-
ence of such
a physical
property.

7. With regard to the concept of '* mass," it is to
be observed that the formulation of Newton, which de-
fines mass to be the quantity of matter of a body as
measured by the product of its volume and density, is
unfortunate. As we can only define density as the mass
of unit of volume, the circle is manifest. Newton felt
distinctly that in every body there was inherent a prop-
erty whereby the amount of its motion was determined
and perceived that this must be different from weight.
He called it, as we still do, mass ; but he did not suc-
ceed in correctly stating this perception. We shall re-
vert later on to this point, and shall stop here only to
make the following preliminary'remarks.

8. Numerous experiences, of which a sufficient num-
ber stood at Newton's disposal, point clearly to the ex-
istence of a property distinct from weight, whereby the

quantity of motion of the
body to which it belongs is
determined. If (Fig. 126)
we tie a fly-wheel to a rope
and attempt to lift it by
means of a pulley, we feel
the weight of the fly-wheel.
If the wheel be placed
on a perfectly cylindrical axle and well balanced, it
will no longer assume by virtue of its weight any de-
terminate position. Nevertheless, we are sensible of

that the.moon is heavy with respect to the earth in the proportion of its mass,
and our seas with respect to the moon ; and all the planets with respect to one
another, and the comets also with respect to the sun ; we must, in conformity
with this rule, declare, that all bodies are heavy with respect to one another.

'* Rule IV. In experimental phsrsics pifepositions collected by induction
from phenomena are to be regarded either as accurately true or very nearly
true, notwithstanding any contrary hypotheses, till other phenomena occur, by
which they are made more accurate, or are rendered subject to exceptions.

"This rule must be adhered to, that the results of induction may not be
annulled by hypotheses."

THE PRINCIPLES OF D YNAMICS. 195

a powerful resistance the moment we endeavor to set Maaa dis-
tinct froin
the wheel in motion or attempt to stop it when in mo- weight.

tion. This is the phenomenon that led to the enuncia-
tion of a distinct property of matter termed inertia, or
'* force" of inertia — a step which, as we have already
seen, and shall further explain below is unnecessary.
Two equal loads simultaneously raised, offer resistance
by their weight. Tied to the extremities of a cord that
passes over a pulley, they offer resistance to any mo-
tion, or rather to any change of velocity of the pulley,
by their mass. A large weight hung as a pendulum
on a very long string can be held at an angle of slight
deviation from the line of equilibrium with very little
effort. The weight-component that forces the pendu-
lum into the position of equilibrium, is very small.
Yet notwithstanding this we shall experience a con-
siderable resistance if we suddenly attempt to move or
stop the weight. A weight that is just supported by a
balloon, although we have no longer to overcome its
gravity, opposes a perceptible resistance to motion.
Add to this the fact that the same body experiences in
different geographical latitudes and in different parts
of space very unequal gravitational accelerations and
we shall clearly recognise that mass exists as a property
wholly distinct from weight determining the amount of
acceleration which a given force communicates to the
body to which it belongs.

9. Important is Newton'^ demonstration that the Mass meas-
mass of a body may, nevertheless, under certain con- weight,
ditions, be measured by its weight. Let us suppose a
body to rest on a support, on which it exerts by its weight
a pressure. The obvious inference is that 2 or 3 such
bodies, or one-half or one-third of such a body, will pro-
duce a corresponding pressure 2, 3, ^, or \ times as

196

TlfE SCIENCE OF MECHANICS.

The prere-
quisites of
the meas-
urement of
mass by
weight.

Newton's
establish-
ment of
these pre-
requisites.

I 1 I

Fig. \vj.

great. If we imagine the acceleration of descent in-
creased, diminished, or wholly removed, we shall ex-
pect that the pressure also will be increased, dimin-
ished, or wholly removed. We thus see^ that the pres-
sure attributable to weight increases, decreases, and

vanishes along with the ** quan-
tity of matter " and the magni-
tude of the acceleration of de-
scent. In the simplest manner
imaginable we conceive the pres-
sure/ as quantitatively representable by the product of
the quantity of matter m into the acceleration of descent
g — by f=zmg. Suppose now we have two bodies that
exert respectively the weight-pressures /, /', to which
we ascribe the " quantities of matter ' ' «r, m\ and which
are subjected to the accelerations of descent g, g'\ then
/ = mg and p' = m g. If, now, we were able to prove,
that, independently of the material (chemical) compo-
sition of bodies, g = g' at every same point on the
earth's surface, we should obtain m/m' =p/p'\ that is
to say, on the same spot of the earth's surface, it would
be possible to measure mass by weight.

Now Newton established this fact, that g is inde-
pendent of the chemical composition of bodies, by
experiments with pendulums of equal lengths but dif-
ferent material, which exhibited equal times of oscilla-
tion. He carefully allowed, in these experiments, for
the disturbances due to the resistance of the air ; this
last factor being eliminated by constructing from differ-
ent materials spherical pendulum-bobs of exactly the
same size, the weights of which were equalised by ap-
propriately hollowing the spheres. Accordingly, all
bodies may be regarded as affected with the same gy and

THE PRINCIPLES OF D YNAMICS. 197

their quantity of matter or mass can, as Newton pointed
out, be measured by their weight.

If we imagine a rigid partition placed between an Supp!e-
assemblage of bodies and a magnet, the bodies, if the considera-
magnet be powerful enough, or at least the majority
of the bodies, will exert a pressure on the partition.
But it would occur to no one to employ this magnetic
pressure, in the manner we employed pressure due to
weight, as a measure of mass. The strikingly notice-
able inequality of the accelerations produced in the
different bodies by the magnet excludes any such idea.
The reader will furthermore remark that this whole
argument possesses an additional dubious feature, in
that the concept of mass which up to this point has
simply been named and feit as a necessity, but not de-
fined, is assumed by it.

10. To Newton we owe the distinct formulation ofihedoc-
the principle of the composition of forces.* If a body composi-
is simultaneously acted on by two forces (Fig. 1 28), forces,
of which one would produce the
motion AB and the other the
motion AC in the same interval
of time, the body, since the two
forces and the motions produced ^**' '*^*

by them are independent of each other, will move in that
interval of time to AD. This conception is in every
respect natural, and distinctly characterises the essen-
tial point involved. It contains none of the artificial
and forced characters that were afterwards imported
into the doctrine of the composition of forces.

We may express the proposition in a somewhat

• Roberval's (x668) achievements with respect to the doctrine of the com-
position of forces are also to be mentioned here. Varignon and Lami have al-
ready bean referred to. (See the text, page 36.)

(

198 THE SCIENCE OF MECHANICS.

Diftcusaion different manner, and thus bring it nearer its modem
trine of the form. The accelerations that different forces impart

compost- _ - , , . , -

tion of to the same body are at the same time the measure of

forces

these forces. But the paths described in equal times
are proportional to the accelerations. Therefore the
latter also may serve as the measure of the forces. We
may say accordingly : If two forces, which are propor-
tional to the lines AB and AC^ act on a body A in the
directions AB and -^ C, a motion will result that could
also be produced by a third force acting alone in the
direction of the diagonal of the parallelogram con-
structed on AB and AC and proportional to that di-
agonal. The latter force, therefore, may be substituted
for the other two. Thus, if q) and ^ are the two ac-
celerations set up in the directions AB and A C, then
for any definite interval of time /, AB = q)t^ /2, AC=-
tl) t^ /2. If, now, we imagine-^/? produced in the same
interval of time by a single force determining the accel-
eration Xi we get

AD = a:^2/2, and AB : AC: AD = (p : tf^ \ x-
As soon as we have perceived the fact that the forces are
independent of each other, the principle of the paral-
lelogram of forces is easily reached from Galileo's no-
tion of force. Without the assumption of this inde-
pendence any effort to arrive abstractly and philosoph-
ically at the principle, is in vain.
The law of II. Perhaps the most important achievement of
reaction. Newton with rcspcct to the principles is the distinct
and general formulation of the law of the equality of
action and reaction, of pressure and counter-pressure.
Questions respecting the motions of bodies that exert
a reciprocal influence on each other, cannot be solved
by Galileo's principles alone. A new principle is ne-
cessary that will define this mutual action. Such a

TFIE PRINCIPLES OF DYNAMICS. 199

principle was that resorted to by Huygens in his inves-
tigation of the centre of oscillation. Such a principle
also is Newton's law of action and reaction.

A body that presses or pulls another body is, ac- Newton's

^. deduction

cording to Newton, pressed or pulled in exactly the of the law

T , , 1 , , ^ 1 of action

same degree by that other body. Pressure and counter- and reac-
pressure, force and counter-force, are always equal to
each other. As the measure of force is defined by
Newton to be the quantity of motion or momentum
(mass X velocity) generated in a unit of time, it conse-
quently follows that bodies that act on each other com-
municate to each other in equal intervals of time equal
and opposite quantities of motion ( momenta), or re-
ceive contrary velocities reciprocally proportional to
their masses.

Now, although Newton's law, in the form here ex- The rela-
tive imme-
pressed, appears much more simple, more immediate, diacy or^

and at first glance more admissible than that of Huy- and Huy-

.„,,,,., . , gens's pria-

gens, It will be found that it by no means contains less cipies.
unanalysed experience or fewer instinctive elements.
Unquestionably the original incitation that prompted
the enunciation of the principle was of a purely instinc-
tive nature. We know that we do not experience any
resistance from a body until we seek to set it in motion.
The more swiftly we endeavor to hurl a heavy . stone
from us, the more our body is forced back by it. Pres-
sure and counter-pressure go hand in hand. The as-
sumption of the equality of pressure and counter- pres-
sure is quite immediate if, using Newton's own illus-
tration, we imagine a rope stretched between two bod-
ies, or a distended or compressed spiral spring between
them.

There exist in the domain of statics very many in-
stinctive perceptions that involve the equality of pres-

200 THE SCIENCE OF MECHANICS.

Statical ex- sure and counter-pressure. The trivial experience that
which point one cauuot lift one's self by pulling on one's chair is
enceof the of this character. In a scholium in which he cites the

Irw

physicists Wren, Huygens, and Wallis as his prede-
cessors in the employment of the principle, Newton
puts forward similar reflections. He imagines the
earth, the single parts of which gravitate towards one
another, divided by a plane. If the pressure of the
one portion on the other were not equal to the counter-
pressure, the earth would be compelled to move in the
direction of the greater pressure. But the motion of
a body can, so far as our experience goes, only be de-
termined by other bodies external to it. Moreover,
we might place the plane of division referred to at any
point we chose, and the direction of the resulting mo-
tion, therefore, could not be exactly determined.
The con- 12. The indistinctness of the concept of mass takes

cept of mass

in its con- a Very palpable form when we attempt to employ the

oection

wi^h this principle of the equality of action and reaction dynam-
ically. Pressure and counter-pressure may be equal.
But whence do we know that equal pressures generate
velocities in the inverse ratio of the masses ? Newton,
indeed, actually felt the necessity of an experimental
corroboration of this principle. He cites in a scholium,
in support of his proposition, Wren*s experiments on
impact, and made independent experiments himself.
He enclosed in one sealed vessel a magnet and in an-
other a piece of iron, placed both in a tub of water,
and left them to their mutual action. The vessels ap-
proached each other, collided, clung together, and af-
terwards remained at rest. This result is proof of the
equality of pressure and counter-pressure and of equal
and opposite momenta (as we shall learn later on,
when we come to discuss the laws of impact).

law.

THE PRINCIPLES OF DYNAMICS. 201

The reader has already felt that the various enunci- The merits

. ^Y . , 11* ^^^ defects

ations of Newton with respect to mass and the prm- of Newton's
ciple of reaction, hang consistently together, and that
they support one another. The experiences that lie at
their foundation are : the instinctive perception of the
connection of pressure and counter-pressure ; the dis-
cernment that bodies offer resistance to change of ve-
locity independently of their weight, but proportion-
ately thereto ; and the observation that bodies of greater
weight receive under equal pressure smaller velocities.
Newton's sense of ivhat fundamental concepts and prin-
ciples were required in mechanics was admirable. The
form of his enunciations, however, as we shall later in-
dicate in detail, leaves much to be desired. But we have
no right to underrate on this account the magnitude of
his achievements ; for the difficulties he had to conquer
were of a formidable kind, and he shunned them less
than any other investigator.

IV.

DISCUSSION AND ILLUSTRATION OF THE PRINCIPLE OF

REACTION.

I. We shall now devote ourselves a moment ex- The princi-
clusively to the Newtonian ideas, and seek to bring the Son?
principle of reaction more clearly home to our mind

/ Fig. 109. Fig. 130.

and feeling. If two masses (Fig. 129) MsLud m act on
one another, they impart to each other, according
to Newton, contrary velocities V and z/, which are in-
versely proportional to their masses, so that

202 THE SCIENCE OF MECHANICS.

General The appearance of greater evidence may be im-

elucidation

of the orin- parted to this principle by the following consideration.

action. We imagine first (Fig. 130) two absolutely equal bodies
a^ also absolutely alike in chemical constitution. We
set these bodies opposite each other and put them in
mutual action ; then, on the supposition that the in-
fluences of any third body and of the spectator are ex-
cluded, the communication of equal and contrary velo-
cities in the direction of the line joining the bodies is
the sole uniquely determined interaction.

Now let us group together in A (Fig. 131) »« such
bodies a, and put at B over against them m' such
bodies a. We have then before us bodies whose quan-

CSX50

Fig. 131. Fig. 13a.

tities of matter or masses bear to each other the pro-
portion m : m'. The distance between the groups we
assume to be so great that we may neglect the exten-
sion of the bodies. Let us regard now the accelera-
tions a, that every two bodies a impart to each other,
as independent of each other. Every part of Ay then,
will receive in consequence of the action of B the ac-
celeration w'ar, and every part of B in consequence of
the action of A the acceleration n^a — accelerations
which will therefore be inversely proportional to the
masses.

2. Let us picture to ourselves now a mass M (Fig.
132) joined by some elastic connection with a mass m,
both masses made up of bodies a equal in all respects.
Let the mass m receive from some external source an
acceleration (p. At once a distortion of the connection
is produced, by which on the one hand m is retarded

THE PRINCIPLES OF D YNAMICS. 203

and on the other M accelerated. When both masses The deduc-
have begun to move with the same acceleration, all notion of
further distortion of the connection ceases. If we call force."
a the acceleration of M and ft the diminution of the
acceleration of w, then a = <p — /?, where agreeably
to what precedes aM =2 fim. From this follows

If we were to enter more exhaustively into the de-
tails of this last occurrence, we should discover that
the two masses, in addition to their motion of progres-
sion, also generally perform with respect to each other
motions of oscillation. If the connection on slight dis-
tortion develop a powerful tension, it will be impos-
sible for any great amplitude of vibration to be reached,
and we may entirely neglect the oscillatory motions,
as we actually have done.

If the expression or = « tp/M + m, which deter-
mines the acceleration of the entire system, be ex-
amined, it will be seen that the product m <p plays a
decisive part in its determination. Newton therefore
invested this product of the mass into the acceleration
imparted to it, with the name of "moving force."
M -\- niy on the other hand, represents the entire mass
of the rigid system. We obtain, accordingly, the accel-
eration of any mass m* on which the
moving force / acts, from the expres-
sion Plin\

3. To reach this result, it is not at

\m,\ \m

Fig. 133.

all necessary that the two connected
masses should act directly on each other in all their
parts. We have, connected together, let us say, the
three masses m^, m^, m,, where m^ is supposed to act

204 THE SCIENCE OF MECHANICS.

A condition only On m^, and m^ only on «r„. Let the mass m. re-

whichdoes . ^ 8 ^ 2 i

not affect ceive ftom some external source the acceleration a?.

the pre- .

vioua re- In the distortion that follows, the

suit.

masses m^ m^ m^

receive the accelerations -^ 6 -{-ft -\- <P

— y — a.

Here all accelerations to the right are reckoned as
positive, those to the left as negative, and it is obvious
that the distortion ceases to increase

when 6 = /3 — y, d = q> — a,
where Sm^ = ym^y am^ =z ftm^.

The resolution of these equations yields the com-
mon acceleration that all the masses receive ; namely,

d =

m^<p

— a result of exactly the same form as before. When
therefore a magnet acts on a piece of iron which is
joined to a piece of wood, we need not trouble our-
selves about ascertaining what particles of the wood
are distorted directly or indirectly (through other par-
ticles of the wood) by the motion of the piece of iron.
The considerations advanced will, in some meas-
ure, perhaps, have contributed towards clearly impress-
ing on us the great importance for mechanics of the
Newtonian enunciations. They will also serve, in a

I subsequent place, to ren-

der more readily obvious

^W^//M<^f^fm the defects of these enun-

ciations.

^/^^&,//^^a.m

\ f 4. Let us now turn to

Fig. 134. a few illustrative physical

examples of the principle of reaction. We consider,
say, a load Z on a table 71 The table is pressed by

THE PRINCIPLES OF D YNAMICS. 205

the load Just so much, and so much only, as it in return Some phys-
presses the load, that is prevents the same from falling, pies of the
If / is the weight, m the mass, and g the acceleration of reaction,
of gravity, then by Newton *s conception /= /«^. If
the table be let fall vertically downwards with the ac-
celeration of free descent g, all pressure on it ceases.
We discover thus, that the pressure on the table is de-
termined by the relative acceleration of the load with
respect to the table. If the table fall or rise with the
acceleration y, the pressure on it is respectively fn(^g —
y) and tn{g -\- y). Be it noted, however, that no
change of the relation is produced by a constant velocity
of ascent or descent. The relative acceleration is de-
terminative.

Galileo knew this relation of things very well. The The pres-
doctrine of the Aristotelians, that bodies of greater parts off aii-
weight fall faster than bodies of less weight, he not only
refuted by experiments, but cornered his adversaries
by logical arguments. Heavy bodies fall faster than
light bodies, the Aristotelians said, because the upper
parts weigh down on the under parts and accelerate
their descent. In that case, returned Galileo, a small
body tied to a larger body must, if it possesses in se the
property of less rapid descent, retard the larger. There-
fore, a larger body falls more slowly than a smaller
body. The entire fundamental assumption is wrong,
Galileo says, because one portion of ^falling body can-
not by its weight under any circumstances press an-
other portion.

A pendulum with the time of oscillation 7^= n V Ijg^ a failing
would acquire, if its axis received the downward accel- p^ ° "™'

eration y^ the time of oscillation 7*= nV l/g y^

and if let fall freely would acquire an infinite time of
oscillation, that is, would cease to oscillate.

2o6

THE SCIENCE OF MECHANICS.

The sensa-
tion of fall'
ing.

PogKen-
dorff's ap-
paratus.

We ourselves, when we jump or fall from an eleva-
tion, experience a peculiar sensation, which must be
due to the discontinuance of the gravitational pressure
of the parts of our body on one another — the blood, and
so forth. A similar sensation, as if the ground were
sinking beneath us, we should have on a smaller planet,
to which we were suddenly transported. The sensation
of constant ascent, like that felt in an earthquake,
would be produced on a larger planet.

5. The conditions referred to are very beautifully
illustrated by an apparatus (Fig. 135^) constructed
by Poggendorff. A string loaded at both extremities

\P-P

VflP^P D

Fig. 135a. FiR- X35b.

by a weight P (Fig. i35<j) is passed over a pulley r,
attached to the end of a scale-beam. A weight / is
laid on one of the weights £rst mentioned and tied by
a fine thread to the axis of the pulley. The pulley
now supports the weight 2 -P -|- /. Burning away the
thread that holds the over-weight, a uniformly accel-
erated motion begins with the acceleration y, with
which P -{- p descends and P rises. The load on the
pulley is thus lessened, as the turning of the scales in-
dicates. The descending weight P is counterbalanced
by the rising weight P, while the added over-weight,
instead of weighing/, now weighs ip/g)ig — y). And
since y = (//2 P -f /) g, we have now to regard the
load on the pulley, not as/, but as/(2 P/2 P-^-p)- The

THE PRINCIPLES OF DYNAMICS. 207

descending weight, only partially impeded in its motion
of descent, exerts only a partial pressure on the pulley.

We may vary the experiment. We pass a thread a vwi
loaded at one extremity with the weight P over the «iper
pulleys a, b, d, of the apparatus as indicated in Fig.

135^., tie the unloaded extremity at m, and equilibrate
the balance. If we pull on the string at m, this can-
not directly affect the balance since the direction of the
string passes exactly through its axis. But the side a
immediately falls. The slackening of the string causes
a to rise. An unacceleraled laaXian of the weights would

2o8

THE SCIENCE OF MECHANICS.

y -■/■-■■/,

* 1
1 I

• 1

I »*
! A

1 /

Fig. 136.

not disturb the equilibrium. But we cannot pass from
rest to motion without acceleration.
Thesuspen- 6. A phenomenon that strikes us at first glance is,

sion of mi- .,,.«• -r

nute bodies that mmute bodies of greater or less specific gravity

different than the liquid m which they are immersed, if sum-
specific .
gravity. ciently small, remain suspended a very long time in the

liquid. We perceive at once that
particles of this kind have to over-
come the friction of the liquid. If the
cube of Fig. 136 be divided into 8
parts by the 3 sections indicated,
and the parts be placed in a row,
their mass and over-weight will re-
main the same, but their cross-sec-
tion and superficial area, with which the friction goes
hand in hand, will be doubled.
Do such Now, the opinion has at times been advanced with

suspended

Darticies af- respect to this phenomenon that suspended particles
specific of the kind described have no influence on the specific
the support- gravity indicated by an areometer immersed in the
liquid, because these particles are themselves areo-
meters. But it will readily be seen that if the sus-
pended particles rise or fall with constant velocity, as
in the case of very small particles immediately occurs,
the effect on the balance and the areometer must be
the same. If we imagine the areometer to oscillate
about its position of equilibrium, it will be evident
that the liquid with all its contents will be moved with
it. Applying the principle of virtual displacements,
therefore, we can be no longer in doubt that the areo-
meter must indicate the mean specific gravity. We
may convince ourselves of the untenability of the rule
by which the areometer is supposed to indicate only
the specific gravity of the liquid and not that of the sus-

THE PRINCIPLES OF D YNAMICS. 209

pended particles, by the following consideration. In a
liquid A a smaller quantity of a heavier liquid B is in-
troduced and distributed in fine drops. The areometer,
let us assume, indicates only the specific gravity of
A, Now, take more and more of the liquid B, finally
just as much of it as we have of -^: we can, then, no
longer say which liquid is suspended in the other, and
which specific gravity, therefore, the areometer must
indicate.

7. A phenomenon of an imposing kind, in which The phe-
the relative acceleration of the bodies concerned is the tides,
seen to be determinative of their mutual pressure, is
that of the tides. We will enter into this subject here
only in so far as it may serve to illustrate the point we
are considering. The connection of the phenomenon
of the tides with the motion of the moon asserts itself
in the coincidence of the tidal and lunar periods, in
the augmentation of the tides at the full and new
moons, in the daily retardation of the tides (by about
50 minutes), corresponding to the retardation of the
culmination of the moon, and so forth. As a matter
of fact, the connection of the two occurrences was very
early thought of. In Newton's time people imagined
to themselves a kind of wave of atmospheric pressure,
by means of which the moon in its motion was sup-
posed to create the tidal wave.

The phenomenon of the tides makes, on every one its impos-
that sees it for the first time in its full proportions, anle?.^ *'**^
overpowering impression. We must not be surprised,
therefore, that it is a subject that has actively engaged
the investigators of all times. The warriors of Alex-
ander the Great had, from their Mediterranean homes,
scarcely the faintest idea of the phenomenon of the
tides, and they were, therefore, not a little taken aback

2IO

THE SCIENCE OF MECIIANTCS.

Extract
from Cur-
tins Rufus.

Describing
the effect
on the army
of Alexan-
der the
Great of the
tides at the
mouth of
the Indus.

by the sight of the powerful ebb and flow at the mouth
of the Indus ; as we learn from the account of Curtius
Rufus {De Rebus Gestis Alexandri Magni), whose
words we here literally, quote :

*'34. Proceeding, now, somewhat more slowly in
' their course, owing to the current of the river being
' slackened by its meeting the waters of the sea, they

< at last reached a second island in the middle of the

* river. Here they brought the vessels to the shore,
*and, landing, dispersed to seek provisions, wholly
'unconscious of the great misfortune that awaited
' them.

*' 35. It was about the third hour, when the ocean,

< in its constant tidal flux and reflux, began to turn

* and press back upon the river. The latter, at first

* merely checked, but then more vehemently repelled,

< at last set back in the opposite direction with a force

< greater than that of a rushing mountain torrent.

* The nature of the ocean was unknown to the multi-

* tude, and grave portents and evidences of the wrath
*of the Gods were seen in what happened. With

* ever- increasing vehemence the sea poured in, com-
' pletely covering the fields which shortly before were
' dry. The vessels were lifted and the entire fleet dis-

* persed before those who had been set on shore, ter-

< rifled and dismayed at this unexpected calamity,
' could return. But the more haste, in times of great
' disturbance, the less speed. Some pushed the ships

* to the shore with poles ; others, not waiting to adjust
' their oars, ran aground. Many, in their great haste
' to get away, had not waited for their companions,

* and were barely able to set in motion the huge, un-

* manageable barks ; while some of the ships were too

* crowded to receive the multitudes that struggled to

THE PRINCIPLES OF DYNAMICS. 211

**get aboard. The unequal division impeded all. TheThedisaa-
** cries of some clamoring to be taken aboard, of others ander's
"crying to put off, and the conflicting commands of
"men, all desirous of different ends, deprived every one
* * of the possibility of seeing or hearing. Even the
"steersmen were powerless; for neither could their
" cries be heard by the struggling masses nor were their
"orders noticed by the terrified and distracted crews.
"The vessels collided, they broke off each other's oars,
" they plunged against one another. One would think
" it was not the fleet of one and the same army that
"was here in motion, but two hostile fleets in combat.
"Prow struck stern; those that had thrown the fore-
"most in confusion were themselves thrown into con-
" fusion by those that followed; and the desperation
"of the struggling mass sometimes culminated in
"hand-to-hand combats.

"36. Already the tide had overflown the fields sur-
" rounding the banks of the river, till only the hillocks
"jutted forth from above the water, like islands.
" These were the point towards which all that had given
"up hope of being taken on the ships, swam. The
"scattered vessels rested in part in deep water, where
"there were depressions in the land, and in part lay
"aground in shallows, according as the waves had
"covered the unequal surface of the country. Then,
"suddenly, a new and greater terror took possession
"of them. The sea began to retreat, and its waters
"flowed back in great long swells, leaving the land
" which shortly before had been immersed by the salt
"waves, uncovered and clear. The ships, thus for-
"saken by the water, fell, some on their prows, some
" on their sides. The fields were strewn with luggage,
"arms, and pieces of broken planks and oars. The

ii
it

212 THE SCIENCE OF MECHANICS.

The dismay " soldicrs dared neither to venture on the land nor to

ofthearmy. • • i i • r i %

' * remain m the ships, for every moment they expected
' ''something new and worse than had yet befallen
•**them. They could scarcely believe that that which
*' they saw had really happened — a shipwreck on dry
"land, an ocean in a river. And of their misfortune
*' there seemed no end. For wholly ignorant that the
tide would shortly bring back the sea and again set
their vessels afloat, they prophesied hunger and dir-
est distress. On the fields horrible animals crept
"about, which the subsiding floods had left behind.
The efforts "37. The night fell, and even the king was sore

of the king "^ o ' o

and the re- "distressed at the slight hope of rescue. But his so-
lum of the o X-

tide. <« licitude could not move his unconquerable spirit. He

"remained during the whole night on the watch, and
" despatched horsemen to the mouth of the river, that,
" as soon as they saw the sea turn and flow back, they
"might return and announce its coming. He also
"commanded that the damaged vessels should be re-
" paired and that those that had been overturned by
"the tide should be set upright, and ordered all to be
"near at hand when the sea should again inundate the
"land. After he had thus passed the entire night in
"watching and in exhortation, the horsemen came
"back at full speed and the tide as quickly followed.
"At first, the approaching waters, creeping in light
"swells beneath the ships, gently raised them, and,
"inundating the fields, soon set the entire fleet in mo-
"tion. The shores resounded with the cheers and
" clappings of the soldiers and sailors, who celebrated
" with immoderate joy their unexpected rescue. *But
" whence, ' they asked, in wonderment, *had the sea
" so suddenly given back these great masses of water?
"Whither had they, on the day previous, retreated?

THE PRINCIPLES OF DYNAMICS, 213

** And what was the nature of this element, which now
"opposed and now obeyed the dominion of the hours? *
*' As the king concluded from what had happened that
'*the fixed time for the return of the tide was after
"sunrise, he set out, in order to anticipate it, at mid-
** night, and proceeding down the river with a few
"ships he passed the mouth and, finding himself at
"last at the goal of. his wishes, sailed out 400 stadia
"into the ocean. He then offered a sacrifice to the
"divinities of the sea, and returned to his fleet."

8. The essential point to be noted in the explication TheexpU-
of the tides is, that the earth as a rigid body can re- the phe-
ceive but one determinate acceleration towards the the tides,
moon, while the mobile particles of water on the sides
nearest to and remotest from the moon can acquire
various accelerations.

Fig. 137.

Let us consider (Fig. 137) on the earth E^ opposite
which stands the moon M^ three points A^ B^ C. The
accelerations of the three points in the direction of the
moon, if we regard them as free points, are respect-
ively (p-\- /J <p, <p, (p — J q). The earth as a whole,
however, has, as a rigid body, the acceleration q). The
acceleration towards the centre of the earth we will
call g. Designating now all accelerations to the left
as negative, and all to the right as positive, we get the
following table :

' L '

2x4 THE SCIENCE OF MECHANICS.

ABC

— (9 + ^<P)^ —9y —i.9 — ^9)

where the symbols of the first and second lines repre-
sent the accelerations which the free points that head
the columns receive, those of the third line the accel-
eration of corresponding rigid points of the earth, and
those of the fourth line, the difference, or the resultant
accelerations of the free points towards the earth. It
will be seen from this result that the weight of the water
at A and C is diminished by exactly the same amount.
The water will rise at A and C (Fig. 137). A tidal
wave will be produced at these points twice every
day.
A variation It is a fact not always sufficiently emphasised, that
nomenon. the phenomenon would be an essentially different one
if the moon and the earth were not affected with ac-
celerated motion towards each other but were relatively
fixed and at rest. If we modify the considerations
presented to comprehend this case, we must put for the
rigid earth in the foregoing computation, ^ = 0 simply.
We then obtain for

the free points .... A C

the accelerations . . — (9> + ^<p\ — {<P — ^9)y

or (g—J<p) — <p, _(^^r__ J^)__^

or g' — <Py —W+<P)y

where g'=g — J (p. In such case, therefore, the
weight of the water at A would be diminished, and the
weight at C increased ; the height of the water at A

THE PRINCIPLES OF D YNAMICS.

215

would be increased, and the height at C diminished.
The water would be elevated only on the side facing
the moon. (Fig. 138.)

Fig. 138.

g. It would hardly be worth while to illustrate An iiiustra-
propositions best reached deductively, by experiments men"***"
that can only be performed with difficulty. But such
experiments are not beyond the limits of possibility.
If we imagine a small iron sphere K to swing as a
conical pendulum about the pole of a
magnet N (Fig. 139), and cover the
sphere with a solution of magnetic sul-
phate .of iron, the fluid drop should, if
the magnet is sufficiently powerful, rep-
resent the phenomenon of the tides. But
if we imagine the sphere to be fixed and
at rest with respect to the pole of the
magnet, the fluid drop will certainly not
be found tapering to a point both on
the side facing and the side opposite to
the pole of the magnet, but will remain suspended only
on the side of the sphere towards the pole of the
magnet.

10. We must not, of course, imagine, that the some fur-
entire tidal wave is produced at once by the action siderations.
o'f the moon. We have rather to conceive the tide
as an oscillatory movement maintained by the moon.
If, for example, we should sweep a fan uniformly and

Fig. 139.

2i6 THE SCIENCE OF MECHANICS.

continuously along over the surface of the water of a
circular canal, a wave of considerable magnitude fol-
lowing in the wake of the fan would by this gentle and
constantly continued impulsion soon be produced. In
like manner the tide is produced. But in the latter
case the occurrence is greatly complicated by the irreg-
ular formation of the continents, by the periodical
variation of the disturbance, and so forth.

CRITICISM OF THE PRINCIPLE OF REACTION AND OF THE

CONCEPT OF MASS. ,

The con- I. Now that the preceding discussions have made

massf us familiar with Newton's ideas, we are sufficiently
prepared to enter on a critical examination of them.
We shall restrict ourselves primarily in this, to the
consideration of the concept of mass and the principle
of reaction. The two cannot, in such an examination,
be separated ; in them is contained the gist of New-
ton's achievement.
Theexpres- 2. In the first place we do not find the expression

8ion"quan- . . ,, , i , • % a • •^

tityof mat- ** quantity of matter ' adapted to explam and elucidate
the concept ot mass, since that expression itself is not
possessed of the requisite clearness. And this is so,
though we go back, as many authors have done, to an
enumeration of the hypothetical atoms. We only com-
plicate, in so doing, indefensible conceptions. If we
place together a number of equal, chemically homo-
geneous bodies, we can, it may be granted, connect
some clear idea with *' quantity of matter," and we per-
ceive, also, that the resistance the bodies offer to mo-
tion increases with this quantity. But the moment we
suppose chemical heterogeneity, the assumption that

THE PRINCIPLES OF DYNAMICS. 217

there is still something that is measurable by the same Newton's
standard, which something we call quantity of matter, of the con-
may be suggested by mechanical experiences, but is an
assumption nevertheless that needs to be justified.
When therefore, with Newton, we make the assump-
tions, respecting pressure due to weight, that/ = mg^
p' = m'gy and put in conformity with such assumptions
///' = mjm'y we have made actual use in the operation
thus performed of the supfosition, yet to be justified,
that different bodies are measurable by the same stand-
ard.

We might, indeed, arbitrarily posit ^ that m/m' =ip/p'',
that is, might define the ratio of mass to be the ratio
of pressure due to weight when g was the same. But
we should then have to substantiate the use that is made
of this notion of mass in the principle of reaction and
in other relations.

<-n — n-> <-^ ^->.

Pig. 140 a. Fig. X40 b.

3. When two bodies (Fig. 140 a), perfectly equal a new form-
in all respects, are placed opposite each other, we ex- the con-

cept«

pect, agreeably to the principle of symmetry, that they
will produce in each other in the direction of their line
of junction equal and opposite accelerations. But if
these bodies exhibit any difference, however slight, of
form, of chemical constitution, or are in any other re-
spects different, the principle of symmetry forsakes us,
unless we assume or know beforehand that sameness of
form or sameness of chemical constitution, or whatever
else the thing in question may be, is not determina-
tive. If, however, mechanical experiences clearly and
indubitably point to the existence in bodies of a special
and distinct property determinative of accelerations^

2i8 THE SCIENCE OF MECHANICS.

nothing stands in the way of our arbitrarily establish-
ing the following definition :
Definition All thost bodies are bodies of equal mass, which^ mu-

masses. tually acting on each other, produce in each other equal
and opposite accelerations.

We have, in this, simply designated, or named, an
actual relation of things. In the general case we pro-
ceed similarly. The bodies A and B receive respec-
tively as the result of their mutual action (Fig. 140 b)
the accelerations — q) and + qjl , where the senses of
the accelerations are indicated by the signs. We say
then, B has q>lq>' times the mass of A. If we take A
as our unity we assign to that body the mass m which im-
parts to A m times the acceleration that A in the reaction
imparts to it. The ratio of the masses is the negative
inverse ratio of the counter-accelerations. That these
accelerations always have opposite signs, that there
are therefore, by our definition, only positive masses,
is a point that experience teaches, and experience alone
Character cau teach. In our concept of mass no theory is in-
nition. ^ volved \ ** quantity of matter ** is wholly unnecessary in
it ; all it contains is the exact establishment, designa-
tion, and denomination of a fact. (Compare Appendix,

II.)

4. One difficulty should not remain unmentioned in
this connection, inasmuch as its removal is absolutely
necessary to the formation of a perfectly clear concept
of mass. We consider a set of bodies, A, B, C, £>. . .,
and compare them all with A as unit.

A, B, c, A b:, F.

I, m, m , m f m , m

We find thus the respective mass- values, 1, m, m\
m". . . . , and so forth. The question now arises, If we

THE PRINCIPLES OF DYNAMICS. 219

select B as our standard of comparison (as our unit), Discnssion

of & diffi~

shall we obtain for C the mass-value m* Im^ and for D cuity in-

, * volved in

the value m" jm, or will perhaps wholly different values the preced-
result ? More simply, the question may be put thus : laOon.
Will two bodies B^ C, which in mutual action with A
have acted as equal masses, also act as equal masses
in mutual action with each other? No A7^V^/ necessity
exists whatsoever, that two masses that are equal to a
third mass should also be equal to each other. For
we are concerned here, not with a mathematical, but
with a physical question. This will be rendered quite
clear by recourse to an analogous relation. We place
by the side of each other the bodies A, B^ C in the
proportions of weight «, ^, c in which they enter into
the chemical combinations AB and A C. There exists,
now, no logical necessity at all for assuming that the
same proportions of weight b^ c of the bodies By C will
also enter into the chemical combination BC, Expe-
rience, however, informs us that they do. If we place
by the side of each other any set of bodies in the pro-
portions of weight in which they combine with the
body Ay they will also unite with each other in the
same proportions of weight. But no one can l^now
this who has not tried it. And this is precisely the case
with the mass-values of bodies.

If we were to assume that the order of combination The order
of the bodies, by which their mass-values are deter- nation not
mined, exerted any influence on the mass-values, the
consequences of such an assumption would, we should
find, lead to conflict with experience. Let us suppose,
for instance (Fig. 141), that we have three elastic
bodies. Ay By C, movable on an absolutely smooth and
rigid ring. We presuppose that A and B in their
mutual relations comport themselves like equal masses

220 THE SCIENCE OF MECHANICS,

and that B and C do the same. We are then also
obliged to assume, if we wish to avoid conflicts with
experience, that C and A in their mutual relations act
like equal masses. If we impart to ^ a velocity, A
will transmit this velocity by impact to B, and B to C
But if C were to act towards A, say, as a greater mass,

A on impact would acquire a greater
velocity than it originally had while
C would still retain a residue of
what it had. With every revolution
in the direction of the hands of a
watch the vis viva of the system
would be increased. If C were the
^**' "*"• smaller mass as compared with A^

reversing the motion would produce the same result.
But a constant increase of vis viva of this kind is at
decided variance with our experience.
The new 5- The Concept of mass when reached in the man-

ma!»Jm-° ner just developed renders unnecessary the special
pHdtfy the enunciation of the principle of reaction. In the con*
reacdon? ° cept of mass and the principle of reaction, as we have
stated in a preceding page, the same fact is twice form-
ulated; which is redundant. If two masses i and 2
act on each other, our very definition of mass asserts
that they impart to each other contrary accelerations
which are to each other respectively as 2:1.

6. The fact that mass can be measured by weighty
where the acceleration of gravity is invariable, can also
be deduced from our definition of mass. We are'
sensible at once of any increase or diminution of a pres-
sure, but this feeling affords us only a very inexact and
indefinite measure of magnitudes of pressure. An
exact, serviceable measure of pressure springs from
the observation that every pressure is replaceable by

THE PRINCIPLES OF D YATAMICS, Z2t

the pressure of a number of like and commensurable it also in-

1 1 J 1 volves the

weights. Every pressure can be counterbalanced by fact that

. mass can be

the pressure of weights of this kind. Let two bodies measured

. .... by weight.

m and m' be respectively affected in opposite directions
with the accelerations ^ and q/, determined by exter-
nal circumstances. And let the bodies be joined by a
string. If equilibrium prevails, the acceleration ^ in .
M and the acceleration q/ in pi' are exactly balanced
by inUraction, For this case, ac-
cordingly, m(p = tn'<p'. When, < — 1^ VffA — >

therefore, <p = <p'f as is the case ^ ^

when the bodies are abandoned

to the acceleration of gravity, we have, in the case
of equilibrium, also /// =: m\ It is obviously imma-
terial whether we make the bodies act on each other
directly by means of a string, or by means of a string
passed over a pulley, or by placing them on the two
pans of a balance. The fact that mass can be meas-
ured by weight is evident from our definition without
recourse or reference to ''quantity of matter.*'

7. As soon therefore as we, our attention being The Kenerai
drawn to the fact by experience, have perceived in bod- this view,
ies the existence of a special property determinative of
accelerations, our task with regard to it ends with the
recognition and unequivocal designation of this fad.
Beyond the recognition of this fact we shall not get,
and every venture beyond it will only be productive of
obscurity. All uneasiness will vanish when once we
have made clear to ourselves that in the concept of
mass no theory of any kind whatever is contained, but
simply a fact of experience. The concept has hitherto
held good. It is very improbable, but not impossible,
^diat it will be shaken in the future, just as the concep-

222 THE SCIENCE OF MECHANICS.

tion of a constant quantity of heat, which also rested
on experience, was modified by new experiences.

VI.
NEWTON'S VIEWS OF TIME, SPACE, AND MOTION.

I. In a scholium which he appends immediately to
his definitions, Newton presents his views regarding
time and space — views which we shall now proceed to
examine more in detail. We shall literally cite, to this
end, only the passages that are absolutely necessary
to the characterisation of Newton's views.
Newton'8 <«So far, my object has been to explain the senses

views of

time, space, <<in which Certain words little known are to be used in

and motion. . -

**the sequel. Time, space, place, and motion, being
'* words well known to everybody, I do not define. Yet
**it is to be remarked, that the vulgar conceive these
• ** quantities only in their relation to sensible objects.
''And hence certain prejudices with respect to them
*' have arisen, to remove which it will be convenient to
'< distinguish them into absolute and relative, true and
''apparent, mathematical and common, respectively.
Absolute, " I. Absolute, true, and mathematical time, of it-

time. " self, and by its own nature, flows uniformly on, with-

"out regard to anything external. It is also called
* ' duration.

"Relative, apparent, and common time, is some
"sensible and external measure of absolute time (dura-
tion), estimated by the motions of bodies, whether
accurate or inequable, and is commonly employed
in place of true time ; as an hour, a day, a month,
'a year. . .

"The natural days, which, commonly, for the pur-
' ' pose of the measurement of time, are held as equal.
' ' are in reality unequal. Astronomers correct this in-

a
n

THE PRINCIPLES OF D YNAMICS. 223

"equality, in order that they may measure by a truer
"time the celestial motions. It may be that there is
"no equable motion, by which time can accurately be
"measured. All motions can be accelerated and re-
"tarded. But the flow of absolute time cannot be
"changed. Duration, or the persistent existence of
" things, is always the same, whether motions be swift
"or slow or null."

2. It would appear as though Newton in the re- Discussion
marks here cited still stood under the influence of the view of

time.

mediaeval philosophy, as though he had grown unfaith-
ful to his resolve to investigate only actual facts. When
we say a thing A changes with the time, we mean sim-
ply that the conditions that determine a thing A depend
on the conditions that determine another thing B, The
vibrations of a pendulum take place in time when its
excursion depends on the position of the earth. Since,
however, in the observation of the pendulum, we are
not under the necessity of taking into account its de-
pendence on the position of the earth, but may com-
pare it with any other thing (the conditions of which
of course also depend on the position of the earth), the
illusory notion easily arises that ail the things with
which we compare it are unessential. Nay, we may,
in attending to the motion of a pendulum, neglect en-
tirely other external things, and find that for every po-
sition of it our thoughts and sensations are different.
Time, accordingly, appears to be some particular and
independent thing, on the progress of which the posi-
tion of the pendulum depends, while the things that
we resort to for comparison and choose at random ap-
pear to play a wholly collateral part. But we must
not forget that all things in the world are connected
with one another and depend on one another, and that

224 ^^^ SCIENCE OF MECHANICS,

General we ourselves and all our thoughts are also a part of
of Uie'con- nature. It is utterly beyond our power to measure the
time° changes of things by time. Quite the contrary, time
is an abstraction, at which we arrive by means of the
changes of things ; made because we are not restricted
to any one definite measure, all being interconnected.
A motion is termed uniform in which equal increments
of space described correspond to equal increments of
space described by some motion with which we form a
comparison, as the rotation of the earth. A motion
may, with respect to another motion, be uniform. But
the question whether a motion is in itself uniform, is
senseless. With just as little justice, also, may we
speak of an *' absolute time" — of a time independent of
change. This absolute time can be measured by com-
parison with no motion ; it has therefore neither a
practical nor a scientific value ; and no one is justified
in saying that he knows aught about it. It is an idle
metaphysical conception.
Further eiu- It would uot be difficult to show from the points of
the idea, view of psychology, history, and the science of lan-
guage (by the names of the chronological divisions),
that we reach our ideas of time in and through the in-
terdependence of things on one another. In these ideas
the profoundest and most universal connection of things
. is expressed. When a motion takes place in time, it
depends on the motion of the earth. This is not refuted
by the fact that mechanical motions can be reversed.
A number of variable quantities may be so related that
one set can suffer a change without the others being
affected by it. Nature behaves like a machine. The
individual parts reciprocally determine one another.
But while in a machine the position of one part de-
termines the position of all the other parts, in nature

THE PRINCIPLES OF DYNAMICS. 225

more complicated relations obtain. These relations are
best represented under the conception of a number,
«, of quantities that satisfy a lesser number, n\ of equa-
tions. Were n = n\ nature would be invariable. Were
n' = n — 1, then with one quantity all the rest would
be controlled. If this latter relation obtained in na-
ture, time could be reversed the moment this had been
accomplished with any one single motion. But the
true state of things is represented by a different rela-
tion between n and n'. The quantities in question are
partially determined by one another ; but they retain
a greater indeterminateness, or freedom, than in the
case last cited. We ourselves feel that we are such a
partially determined, partially undetermined element
of nature. In so far as a portion only of the changes
of nature depends on us and can be reversed by us,
does time appear to us irreversible, and the time that
is past as irrevocably gone.

We arrive at the idea of time, — to express it briefly Some psy-
and popularly, — by the connection of that which isconaiSera-
contained in the province of our memory with that
which is contained in the province of our sense-percep-
tion. When we say that time flows on in a definite di-
rection or sense, we mean that physical events gene-
rally (and therefore also physiological events) take
place only in a definite sense.* Differences of tem-
perature, electrical differences, differences of level gen-
erally, if left to themselves, all grow less and not
greater. If we contemplate two bodies of different
temperatures, put in contact and left wholly to them-
selves, we shall find that it is possible only for greater
differences of temperature in the field of memory to

* Investigations concerning the physiological nature of the sensations of
dme and space are here excluded from consideration.

226 THE SCIENCE OF MECHANICS,

exist with lesser ones in the field of sense-perception,
and not the reverse. In all this there is simply ex-
pressed a peculiar and profound connection of things.
To demand at the present time a full elucidation of this
matter, is to anticipate, in the manner of speculative
philosophy, the results of all future special investiga-
tion, that is a perfect physical science. (Compare Ap-
pendix, III.)
Newton's 3. Views similar to those concerning time, are de-

views of

space and veloped by Newtou with respect to space and motion.
We extract here a few passages which characterise his
position.

" II. Absolute space, in its own nature and with-
*'out regard to anything external, always remains sim-
**ilar and immovable.

** Relative space is some movable dimension or
"measure of absolute space, which our senses deter-
'*mine by its position with respect to other bodies,
'*and which is commonly taken for immovable [abso-
"lute] space. . . .

** IV. Absolute motion is the translation of a body
"from one absolute place* to another absolute place ;
*' and relative motion, the translation from one relative
" place to another relative place. . . .
Passages " . . . . And thus we use, in common affairs, instead

works. "of absolute places and motions, relathe ones; and
"that without any inconvenience. But in physical
"disquisitions, we should abstract from the senses.
" For it may be that there is no body really at rest, to
"which the places and motions of others can be re-
"ferred. ...

" The effects by which absolute and relative motions

*The place, or locus of a body, according to Newton, is not its position,
but the/aW ofspact which it occupies. It is either absolute or relative. — Tran*.

THE PRINCIPLES OF D YNAMICS.

227

are distinguished from one another, are centrifugal
forces, or those forces in circular motion which pro-
duce a tendency of recession from the axis. For in
a circular motion which is purely relative no such
forces exist ; but in a true and absolute circular mo-
tion they do exist, and are greater or less according
to the quantity of the [absolute] motion.

*' For instance. If a bucket, suspended by a long The rota-
cord, is so often turned about that finally the cord is
strongly twisted, then is filled with water, and held
at rest together with the water ; and afterwards by
the action of a second force, it is suddenly set whirl-
ing about the contrary way, and continues, while the
cord is untwisting itself, for some time in this mo-
tion ; the surface of the water will at first be level,
just as it was before the vessel began to move ; but,
subsequently, the vessel, by gradually communicat-
ing its motion to the water, will make it begin sens-
ibly to rotate, and the water will recede little by little
from the middle and rise up at the sides of the ves-
sel, its surface assuming a concave form. (This ex-
periment I have made myself.)

*'.... At first, when the relative motion of the wa- Relative
ter in the vessel was greatest^ that motion produced motion,
no tendency whatever of recession from the axis ; the
water made no endeavor to move towards the cir-
cumference, by rising at the sides of the vessel, but
remained level, and for that reason its true circular
motion had not yet begun. But afterwards, when
the relative motion of the water had decreased, the
rising of the water at the sides of the vessel indicated
an endeavor to recede from the axis ; and this en-
deavor revealed the real circular motion of the water,
continually increasing, till it had reached its greatest

228

THE SCIENCE OF MECHANICS.

Newton's
criteria for
distinguish-
ing absolute
from rela-
tive motion.

point, when relatively the water was at rest in the
vessel ....

" It is indeed a matter of great difficulty to discover
and effectually to distinguish the true from the ap-
parent motions of particular bodies ; for the parts of
that immovable space in which bodies actually move,
do not come under the observation of our senses.

** Yet the case, is not altogether desperate ; for there
exist to guide us certain marks, abstracted partly
from the apparent motions, which are the differences
of the true motions, and partly from the forces that
are the causes and effects of the true motions. If,
for instance, two globes, kept at a fixed distance
from one another by means of a cord that connects
them, be revolved about their common centre of
gravity, one might, from the simple tension of the
cord, discover the tendency of t^ie globes to recede
from the axis of their motion, and on this basis the
quantity of their circular motion might be computed.
And if any equal forces should be simultaneously
impressed on alternate faces of the globes to augment
or diminish their circular motion, we might, from
the increase or decrease of the tension of the cord,
deduce the increment or decrement of their motion ;
and it might also be found thence on what faces
forces would have to be impressed, in order that the
motion of the globes should be most augmented ;
that is, their rear faces, or those which, in the cir-
cular motion, follow. But as soon as we knew which
faces followed, and consequently which preceded, we
should likewise know the direction of the motion.
In this way we might find both the quantity and the
direction of the circular, motion, considered even in
an immense vacuum, where there was nothing ex-

THE PRINCIPLES OF D YNAMICS. 229

'* temal or sensible with which the globes could be
** compared ....**

4. It is scarcely necessary to remark that in the re- The predi-
flections here presented Newton has again acted con- Newton

, . -..,.. , are not the

trary to his expressed mtention only to mvestigate actual cv^xessAon

_ _ . J • 1 • t o' actual

facts. No one is competent to predicate things about facts.
absolute space and absolute motion ; they are pure
things of thought, pure mental constructs, that cannot
be pr<5duced in experience. All our principles of me-
chanics are, as we have shown in detail, experimental
knowledge concerning the relative positions and mo-
tions of bodies. Even in the provinces in which they
are now recognised as valid, they could not, and were
not, admitted without previously being subjected to
experimental tests. No one is warranted in extending
these principles beyond the boundaries of experience.
In fact, such an extension is meaningless, as no one
possesses the requisite knowledge to make use of it.

Let us look at the matter in detail. When we say that Detailed
a body K alters its direction and velocity solely through matter,
the influence of another body K\ we have asserted
a conception that it is impossible to come at unless
other bodies A, B, C . . . are present with reference
to which the motion of the body K has been estimated.
In reality, therefore, we are simply cognisant of a re-
lation of the body K to A, B, C , , . . If now we sud-
denly neglect A, B, C . . . and attempt to speak of
the deportment of the body JC in absolute space, we
implicate ourselves in a twofold error. In the first
place, we cannot know how A!" would act in the ab-
sence of ^, ^, C . . . ; and in the second place, every
means would be wanting of forming a judgment of the
behaviour of AT and of putting to the test what we had

230 THE SCIENCE OF MECHANICS.

predicated, — which latter therefore would be bereft of

all scientific significance.
The part Two bodies K and K\ which gravitate toward each

bodies of other, impart to each other in the direction of their
inthede- line of junction accelerations inversely proportional to

termination . #ti' ... ..,

of motion, their masses m, m , In this proposition is contained,
not only a relation of the bodies K and K* to otie an-
other, but also a relation of them to other bodies. For
the proposition asserts, not only that K and K* suffer
with respect to one another the acceleration designated
by K (ni -|- m' /r^)y but also that .^experiences the ac-
celeration — Km* jr^ and K* the acceleration + Km/r^
in the direction of the line of junction ; facts which can
be ascertained only by the presence of other bodies.

The motion of a body K can only be estimated by
reference to other bodies A, B, C . . . But since we
always have at our disposal a su£Scient number of
bodies, that are as respects each other relatively fixed,
or only slowly change their positions, we are, in such
reference, restricted to no one definite body and can
alternately leave out of account now this one and now
that one. In this way the conviction arose that these
bodies are indifferent generally.

T*»e hy- It might be, indeed, that the isolated bodies A. B.

pothesisof o > > 97

a medium C . . . play merely a collateral role in the determina-

in space de- . * "^ "^

terminative tiou of the motiou of the body Kt and that this motion

of motion. . ■' ,

is determined by a tnedium in which K exists. In such
a case we should have to substitute this medium for
Newton's absolute space. Newton certainly did not
entertain this idea. Moreover, it is easily demonstrable
that the atmosphere is not this motion-determinative
medium. We should, therefore, have to picture to
ourselves some other medium, filling, say, all space,
with respect to the constitution of which and its kinetic

THE PRINCIPLES OF DYNAMICS, i^t

relations to the bodies placed in it we have at present
no adequate knowledge. In itself such a state of things
would not belong to the impossibilities. It is known,
from recent hydrodynamical investigations, that a rigid
body experiences resistance in a frictionless fluid only
when its velocity changes. True, this result is derived
theoretically from the notion of inertia ; but it might,
conversely, also be regarded as the primitive fact from
which we have to start. Although, practically, and at
present, nothing is to be accomplished with this con-
ception, we might still hope to learn more in the future
concerning this hypothetical medium ; and from the
point of view of science it would be in every respect
a more valuable acquisition than the forlorn idea of
absolute space. When we reflect that we cannot abol-
ish the isolated bodies A, B, C . . ., that is, cannot
determine by experiment whether the part they play is
fundamental or collateral, that hitherto they have been
the sole and only competent means of the orientation
of motions and of the description of mechanical facts,
it will be found expedient provisionally to regard all
motions as determined by these bodies.

5. Let us now examine the point on which New- critical
ton, apparently with sound reasons, rests his distinc- tion of
tion of absolute and relative motion. If the earth is distinction
affected with an absolute rotation about its axis, cen- from reia-

• e % c • 1 1 • tive motion.

tnfugal forces are set up m the earth : it assumes an
oblate form, the acceleration of gravity is diminished
at the equator, the plane of Foucault's pendulum ro-
tates, and so on. All these phenomena disappear if
the earth is at rest and the other heavenly bodies are
affected with absolute motion round it, such that the
same relative rotation is produced. This is, indeed, the
case, if we start ab initio from the idea of absolute space.

232 THE SCIENCE OF MECHANICS.

But if we take our stand on the basis of facts, we shall
find we have knowledge only of relative spaces and mo-
tions. Relatively, not considering the unknown and
neglected medium of space, the motions of the uni-
verse are the same whether we adopt the Ptolemaic or
the Copernican mode of view. Both views are, indeed,
equally correct \ only the latter is more simple and more
practical. The universe is not twice given, with an
earth at rest and an earth in motion ; but only once,
with its relative motions, alone determinable. It is,
accordingly, not permitted us to say how things would
be if the earth did not rotate. We may interpret the
one case that is given us, in different ways. If, how-
ever, we so interpret it that we come into conflict with
experience, our interpretation is simply wrong. The
principles of mechanics can, indeed, be so conceived,
that even for relative rotations centrifugal forces arise,
interpreta- Newton's experiment with the rotating vessel of
experiment Water simply iuforms us, that the relative rotation of
rotatinf! the Water with respect to the sides of the vessel pro-
water, duces no noticeable centrifugal forces, but that such
forces are produced by its relative rotation with respect
to the mass of the earth and the other celestial bodies.
No one is competent to say how the experiment would
turn out if the sides of the vessel increased in thickness
and mass till they were ultimately several leagues thick.
The one experiment only lies before us, and our busi-
ness is, to bring it into accord with the other facts
known to us, and not with the arbitrary fictions of our
imagination.

6. We can have no doubts concerning the signifi-
cance of the law of inertia if we bear in mind the man-
ner in which it was reached. To begin with, Galileo
discovered the constancy of the velocity and direction

THE PRINCIPLES OF D YNAMICS, 233

of a body referred to terrestrial objects. Most terres- The law ot
trial motions are of such brief duration and extent, that the ii^ht of

this view

it is wholly unnecessary to take into account the earth's
rotation and the changes of its progressive velocity with
respect to the celestial bodies. This consideration is
found necessary only in the case of projectiles cast
great distances, in the case of the vibrations of Fou-
cault's pendulum, and in similar instances. When now
Newton sought to apply the mechanical principles dis-
covered since Galileo's time to the planetary system,
he found that, so far as it is possible to form any es-
timate at all thereof, the planets, irrespectively of dy-
namic effects, appear to preserve their direction and
velocity with respect to bodies of the universe that are
very remote and as regards each other apparently fixed,
the same as bodies moving on the earth do with re-
spect to the fixed objects of the earth. The comport-
ment of terrestrial bodies with respect to the earth is
• reducible to the comportment of the earth with respect
to the remote heavenly bodies. If we were to assert
that we knew more of moving objects than this their
last - mentioned, experimentally - given comportment
with respect to the celestial bodies, we should render
ourselves culpable of ^, falsity. When, accordingly, we
say, that a body preserves unchanged its direction and
velocity in space, our ■ assertion is nothing more or less
than an abbreviated reference to the entire universe.
The use of such an abbreviated expression is permit-
ted the original author of the principle, because he
knows, that as things are no difficulties stand in the
way of carrying out its implied directions. But no
remedy lies in his power, if difficulties of the kind men-
tioned present themselves; if, for example, the re-
quisite, relatively fixed bodies are wanting.

234 THE SCIENCE OF MECHANICS.

The reia- 7. Instead, now, of referring a moving body K to

bodies of spacc, that IS to say to a system of coordmates, let us

the uni-

verse to view directly its relation to the bodies of the universe,

each other. . , . , , , r ,- «

by which alone such a system of coordmates can be
determined. Bodies very remote from each other, mov-
ing with constant direction and velocity with respect
to other distant fixed bodies, change their mutual dis-
tances proportionately to the time. We may also say.
All very remote bodies — all mutual or other forces ne-
glected— alter their mutual distances proportionately
to those distances. Two bodies, which, situated at a
short distance from one another, move with constant
direction and velocity with respect to other fixed bod-
ies, exhibit more complicated relations. If we should
regard the two bodies as dependent on one another,
and call r the distance, / the time, and a a constant
dependent on the directions and velocities, the formula
would be obtained: d^rjdt^ = (1/r) [a^ — (//r////)2].
It is manifestly much simpler and clearer to regard the
two bodies as independent of each other and to con-
sider the constancy of their direction and velocity with
respect to other bodies.

Instead of saying, the direction and velocity of a
mass ji in space remain constant, we may also employ
the expression, the mean acceleration of the mass /i
with respect to the masses w, ni\ m". ... at the dis-
tances r, r', r". ... is = 0, or d^{'2mr/2'm)/dt^ = 0.
The latter expression is equivalent to the former, as
soon as we take into consideration a sufficient number
of sufficiently distant and sufficiently large masses.
The mutual influence of more proximate small masses,
which are apparently not concerned about each other,
is eliminated of itself. That the constancy of direction
and velocity is given by the condition adduced, will be

THE PRINCIPLES OF DYNAMICS, 235

seen at once if we construct through u as vertex cones The ezprea-

sion of the

that cut out different portions of space, and set up the uw of iner-
condition with respect to the masses of these separate of this re-
portions. We may put, indeed, for the entire space
encompassing //, d^ {2mr/2tn) /dt^ r=z{i. But the
equation in this case asserts nothing with respect to the
motion of /i, since it holds good for all species of mo-
tion where /i is uniformly surrounded by an infinite
number of masses. If two masses pi^, M2 ^^^^^ o" each
other a force which is dependent on their distance r,
then d^rjdf^ = (/ij + /^2)/('')- ^^^> ^^ ^^® ^afne time,
the acceleration of the centre of gravity of the two
masses or the mean acceleration of the mass-system
with respect to the masses of the universe (by the prin-
ciple of reaction) remains == 0 ; that is to say,

d^
~d~t^

2mr^ 2mr^

^1 ~1S^ + ^» 2m

= 0.

When we reflect that the time-factor that enters The necm-
into the acceleration is nothing more than a quantity ence of a
that is the measure of the distances (or angles of rota- Uon of the

All

tion) of the bodies of the universe, we see that even in
the simplest case, in which apparently we deal with
the mutual action of only two masses, the neglecting
of the rest of the world is impossible. Nature does not
begin with elements, as we are obliged to begin with
them. It is certainly fortunate for us, that we can,
from time to time, turn aside our eyes from the over-
powering unity of the All, and allow them to rest on
individual details. But we should not omit, ultimately
to complete and correct our views by a thorough con-
sideration of the things which for the time being we
left out of account.

8. The considerations just presented show, that it

236

THE SCIENCE OF MECHANICS,

The law of IS not necessarv to refer the law of inertia to a special

inertia does ^>. •% • • • j i

not involve absolute space. On the contrary, it is perceived that

absolute

space. the masses that in the common phraseology exert forces
on each other as well as those that exert none, stand
with respect to acceleration in quite similar relations.
We may, indeed, regard all masses as related to each
other. That accelerations play a prominent part in the
relations of the masses, must be accepted as a fact of
experience ; which does not, however, exclude attempts
to elucidate this fact by a comparison of it with other
facts, involving the discovery of new points of view.
In all the processes of nature the differences of certain

quantities u play a de-
terminative role. Differ-
ences of temperature, of
potential function, and so
forth, induce the natural
processes, which consist
in the equalisation of
these differences. The familiar expressions d^u/tlx^y
d^u/dy^y d^u/dz^y which are determinative of the
character of the equalisation, may be regarded as the
measure of the departure of the condition of any point
from the mean of the conditions of its environment —
to which mean the point tends. The accelerations of
masses may be analogously conceived. The great dis-
tances between masses that stand in no especial force-
relation to one another, change proportionately to each
other. If we lay off, therefore, a certain distance p as
abscissa, and another r as ordinate, we obtain a straight
line. (Fig. 143.) Every r-ordinate corresponding to
a definite p- value represents, accordingly, the mean of
the adjacent ordinates. If a force-relation exists be-
tween the bodies, some value d^r/dt^ is determined

F»« 143.

Natural
processes
consist in
the equali-
sation of
the differ-
ences of
quantities.

THE PRINCIPLES OF D YNAMICS. 237

by it which conformably to the remarks above we may
replace by an expression of the form d^r/dp^. By the
force-relation, therefore, a departure of the r-ordinate
from the mean of the adjacent ordinates is produced,
which would not exist if the supposed force-relation
did not obtain. This intimation will suffice here.

9. We have attempted in the foregoing to give the character
law of inertia a different expression from that in ordi- expression
nary use. This expression will, so long as a suffi- of inertia,
cient number of bodies are apparently fixed in space,
accomplish the same as the ordinary one. It is as
easily applied, and it encounters the same difficulties.
In the one case we are unable to come at an absolute
space, in the other a limited number of masses only is
within the reach of our knowledge, and the summation
indicated can consequently not be fully carried out. It
is impossible to say whether the new expression would
still represent the true condition of things if the stars
were to perform rapid movements among one another.
The general experience cannot be constructed from the
particular case given us. We must, on the contrary,
wait until such an experience presents itself. Perhaps
when our physico-astronomical knowledge has been
extended, it will be offered somewhere in celestial
space, where more violent and complicated motions
take place than in our environment. The most impor- The sim-
tant*result of our reflexions is, however, that precisely c\^\e&ot

mechanics

the apparently simplest mechanical principles are of a tfery are of a
complicated character , that these principles are founded on plfcat^edna-
uncompleted experiences^ nay on experiences that never can all derived
be fully completed, that practically y indeed^ they are suf rience.
ficiently secured^ in view of the tolerable stability of our
environment y to serve as the foundation of mathematical
deduction^ but that they can by no means themselves be re-

238 TJIE SCIENCE OF MECHANICS,

garded as mathematically established truths but only as
principles that not only admit of constant control by expe-
rience but actually require it. This perception is valu-
able in that it is propitious to the advancement of
science. (Compare Appendix, IV.)

VII.
SYNOPTICAL CRITIQUE OF THE NEWTONIAN ENUNCIATIONS.

Newton's I. Now that we have discussed the details with

sufficient particularity, we may pass again under re-
view the form and the disposition of the Newtonian
enunciations. . Newton premises to his work several
definitions, following which he gives the laws of mo-
tion. We shall take up the former first.

Mass. ^^ Definition I, The quantity of any matter is the

** measure of it by its density and volume conjointly.
"... This quantity is what I shall understand by the
** term mass or body in the discussions to follow. It is
* ' ascertainable from the weight of the body in ques-
'*tion. For I have found, by pendulum -experiments
"of high precision, that the mass of a body is propor-
" tional to its weight ; as will hereafter be shown.

Quantity of ^^ Definition II, Quantity of motion is the measure

motion, e • \ % « • i • r

inertia, " of it by the velocity and quantity of matter con-
force, and . . ,
accelera- * ' J Ol n tly .

'* Definition III The resident force [vis insita^ i. e.
"the inertia] of matter is a power of resisting, by
"which every body, so far as in it lies, perseveres in
"its state of rest or of uniform motion in a straight
"line.

Definition IV, An impressed force is any action
upon a body which changes, or tends to change^ its
state of rest, or of uniform motion in a straight line.

tion.

it
it
ti

THE PRINCIPLES OF D YNAMICS.

239

** Definition V. A centripetal force is any force by
which bodies are drawn or impelled towards, or tend
in any way to reach, some point as centre.

^^ Definition VI, The absolute quantity of a centri-
petal force is a measure of it increasing and dimin-
ishing with the efficacy of the cause that propagates
it from the centre through the space round about.

^^ Definition VII. The accelerative quantity of a
centripetal force is the measure of it proportional to
the velocity which it generates in a given time.

^^ Definition VIII. The moving quantity of a cen-
tripetal force is the measured of it proportional to the
motion [See Def. 11.] which it generates in a given
time.

<< The three quantities or measures of force thus dis-
tinguished, may, for brevity's sake, be called abso-
lute, accelerative, and moving forces, being, for dis-
tinction's sake, respectively referred to the centre of
force, to the places of the bodies, and to the bodies
that tend to the centre : that is to say, I refer moving
force to the body, as being an endeavor of the whole
towards the centre, arising from the collective en-
deavors of the several parts ; accelerative force to the
place of the body, as being a sort of efficacy originat-
ing in the centre and diffused throughout all the sev-
eral places round about, in moving the bodies that
are at these places ; and absolute force to the centre,
as invested with some cause, without which moving
forces would not be propagated through the space
round about ; whether this latter cause be some cen-
tral body, (such as is a loadstone in a centre of mag-
netic force, or the earth in the centre of the force of
gravity,) or anything else not visible. This, at least,
is the mathematical conception of forces ; for their

Forces clas-
sified as ab-
solute, ac-
celerative,
and mov-
ing.

The rela-
tions of the
forces thus
distin-
guished.

240

THE SCIENCE OF MECHANICS.

The dis-
tinction
mathemat-
ical and not
physical.

' physical causes and seats I do not in this place con-
' sider.

"Accelerating force, therefore, is to moving force,

* as velocity is to quantity of motion. For quantity

* of motion arises from the velocity and the quantity
' of matter ; and moving force arises from the accel-

* erating force and the same quantity of matter ; the
' sum of the effects of the accelerative force on the sev-

* eral particles of the body being the motive force of
*the whole. Hence, near the surface of the earth,
' where the accelerative gravity or gravitating force is

* in all bodies the same, the motive force of gravity or
'the weight is as the body [mass]. But if we ascend
*to higher regions, where the accelerative force of

* gravity is less, the weight will be equally diminished,
' always remaining proportional conjointly to the mass

* and the accelerative force of gravity. Thus, in those

* regions where the accelerative force of gravity is half

< as great, the weight of a body will be diminished by

* one- half. Further, I apply the terms accelerative and

< motive in one and the same sense to attractions and

* to impulses. I employ the expressions attraction, im-

* pulse, or propensity of any kind towards a centre,

< promiscuously and indifferently, the one for the other;
' considering those forces not in a physical sense, but
'mathematically. The reader, therefore, must not
' infer from any expressions of this kind that I may

< use, that I take upon me to explain the kind or the

< mode of an action, or the causes or the physical rea-
' son thereof, or that I attribute forces in a true or

* physical sense, to centres (which are only mathemat-
' ical points), when at any time I happen to say that
'centres attract or that central forces are in action."

THE PRINCIPLES OF DYNAMICS, 241

2. Definition i is, as has already been set forth, a criticism of
pseudo-definition. The concept of mass is not made Definitions.
clearer by describing mass as the product of the volume

into the density, as density itself denotes simply the
mass of unit of volume. The true definition of mass
can be deduced only from the dynamical relations of
bodies.

To Definition 11, which simply enunciates a mode
of computation, no objection is to be made. Defini-
tion III (inertia), however, is rendered superfluous by
Definitions iv-viii of force, inertia being included and
given in the fact that forces are accelerative.

Definition iv defines force as the cause of the accel-
eration, or tendency to acceleration, of a body. The
latter part of this is justified by the fact that in the
cases also in which accelerations cannot take place,
other attractions that answer thereto, as the compres-
sion and distension etc. of bodies occur. The cause
of an acceleration towards a definite centre is defined
in Definition v as centripetal force, and is distinguished
in VI, VII, and viii as absolute, accelerative, and mo-
tive. It is, we may say, a matter of taste and of form
whether we shall embody the explication of the idea
of force in one or in several definitions. In point of
principle the Newtonian definitions are open to no ob-
jections.

3. The Axioms or Laws of Motion then follow, of Newton's

Lftws of

which Newton enunciates three : Motion.

'* Law /. Every body perseveres in its state of rest
"or of uniform motion in a straight line, except in so
far as it is compelled to change that state by im-
pressed forces."
'* Law II, Change of motion [i. e. of momentum] is
** proportional to the moving force impressed, and takes

242 THE SCIENCE OF MECHANICS,

•

" place in the direction of the straight line in which
'*such force is impressed."
^ ^^ Law III, Reaction is always equal and opposite

<'to action; that is to say, the actions of two bodies
< < upon each other are always equal and directly op-
"posite."

Newton appends to these three laws a number of
Corollaries. The first and second relate to the prin-
ciple of the parallelogram of forces ; the third to the
quantity of motion generated in the mutual action of
bodies ; the fourth to the fact that the motion of the
centre of gravity is not changed by the mutual action
of bodies \ the fifth and sixth to relative motion.
Criticism of 4. We readily perceive that Laws i and ii are con-

f^ Aorf Qfl * Q

laws of tained in the definitions of force that precede. Ac-

motion.

cording to the latter, without force there is no accel-
eration, consequently only rest or uniform motion in a
straight line. Furthermore, it is wholly unnecessary
tautology, after having established acceleration as the
measure of force, to say again that change of motion is
proportional to the force. It would have been enough
to say that the definitions premised were not arbitrary
mathematical ones, but correspond to properties of
bodies experimentally given. The third law apparently
contains something new. But we have seen that it is
unintelligible without the correct idea of mass, which
idea, being itself obtained only from dynamical expe-
rience, renders the law unnecessary.
Thccoroi- The first corollary really does contain something/
these laws. new. But it regards the accelerations determined in)
a body K by different bodies M^ Ny P as self- evidently^,
independent of each other, whereas this is precisely^
what should have been explicitly recognised as a fad
of experience. Corollary Second is a simple applica-

THE PRINCIPLES OF D YNAMICS. 243

tion of the law enunciated in corollary First. The re-
maining corollaries, likewise, are simple deductions,
that is, mathematical consequences, from the concep-
tions and laws that precede.

5. Even if we adhere absolutely to the Newtonian
points of view, and disregard the complications and in-
definite features mentioned, which are not removed
but merely concealed by the abbreviated designations
"Time" and "Space," it is possible to replace New-
ton's enunciations by much more simple, methodically
better arranged, and more satisfactory propositions.
Such, in our estimation, would be the following :

a. Experimental Proposition, Bodies set opposite Proposed
each other induce in each other, under certain circum- tions for
stances to be specified by experimental physics, con- tonian laws
trary accelerations in the direction of their line of junc- tions.
tion. (The principle of inertia is included in this.)

b. Definition, The mass-ratio of any two bodies is
the negative inverse ratio of the mutually induced ac-
celerations of those bodies.

c. Experimental Proposition, The mass-ratios of
bodies are independent of the character of the physical
states (of the bodies) that condition the mutual accel-
erations produced, be those states electrical, magnetic,
or what not ; and they remain, moreover, the same,
whether they are mediately or immediately arrived at.

d. Experimental Proposition. The accelerations
which any number of bodies Ay B, C , . . . induce in a
body K, are independent of each other. (The principle
of the parallelogram of forces follows immediately from
this.)

e. Definition. Moving force is the product of the
mass- value of a body into the acceleration induced in
that body.

244 THE SCIENCE OF MECHANICS.

Extent and Then the remaining arbitrary definitions of the al-

ch&ractcr

of the pro- gebraical expressions "momentum," "vis viva," and
stitutions. the like, might follow. But these are by no means in-
dispensable. The propositions above set forth satisfy
the requirements of simplicity and parsimony which,
on economico-sclentific grounds, must be exacted of
them. They are, moreover, obvious and clear ; for no
doubt can exist with respect to any one of them either
concerning its meaning or its source ; and we always
know whether it asserts an experience or an arbitrary
convention.
The 6. Upon the whole, we may say, that Newton dis-

achieve-

ments of cemed in an admirable manner the concepts and princi-
from the ples that Were sufficiently assured to allow of being fur-
view of bis ther built upon. It is possible that to some extent he
was forced by the difi&culty and novelty of his subject,
in the minds of the contemporary world, to great am-
plitude, and, therefore, to a certain disconnectedness
of presentation, in consequence of which one and the
same property of mechanical processes appears several
times formulated. To some extent, however, he was,
as it is possible to prove, not perfectly clear himself
concerning the import and especially concerning the
source of his principles. This cannot, however, ob-
scure in the slightest his intellectual greatness. He
that has to acquire a new point of view naturally can-
not possess it so securely from the beginning as they
that receive it unlaboriously from him. He has done
enough if he has discovered truths on which future
generations can further build. For every new infer-
ence therefrom affords at once a new insight, a new
control, an extension of our prospect, and a clarifica-
tion of our field of view. Like the commander of an
army, a great discoverer cannot stop to institute petty

THE PRINCIPLES OF D YNAMICS. 245

inquiries regarding the right by which he holds each The
post of vantage he has won. The magnitude of the ments of
problem to be solved leaves no time for this. But at the light of
a later period, the case is different. Newton might re«earch.
well have expected of the two centuries to follow that
they should further examine and confirm the founda-
tions of his work, and that, when times of greater scien-
tific tranquillity should come, the principles of the sub-
ject might acquire an even higher philosophical in-
terest than all that is deducible from them. Then prob-
lems arise like those just treated of, to the solution of
which, perhaps, a small contribution has here been
made. We join with the eminent physicists Thomson
and Tait, in our reverence and admiration of Newton.
But we can only comprehend with difficulty their opin-
ion that the Newtonian doctrines still remain the best
and most philosophical foundation of the science that
can be given.

VIII.

RETROSPECT OF THE DEVELOPMENT OF DYNAMICS.

•

I. If we pass in review the period in which the de- The chief
velopment of dynamics fell, — a period inaugurated by discovery
Galileo, continued by Huygens, and brought to a close fact,
by Newton, — its main result will be found to be the
perception, that bodies mutually determine in each
other accelerations dependent on definite spatial and.
material circumstances, and that there are masses. The
reason the perception of these facts was embodied in
so great a number of principles is wholly an historical
one ; the perception was not reached at once, but slowly
and by degrees. In reality only one great fact was es-
tablished. Different pairs of bodies determine, inde-
pendently of each other, and mutually, in themselves.

246 THE SCIENCE OF MECHANICS,

pairs of accelerations, whose terms exhibit a constant
ratio, the criterion and characteristic of each pair.
This fact Not even men of the calibre of Galileo, Huygens,

greatest in- and Newton were able to perceive this fact at once,
could per- Even they could only discover it piece by piece, as it
in frag- is expressed in the law of falling bodies, in the special

Dients.

law of inertia, in the principle of the parallelogram of
forces, in the concept of mass, and so forth. To-day,
no difi&culty any longer exists in apprehending the unity
of the whole fact. The practical demands of communi-
cation alone can justify its piecemeal presentation in
several distinct principles, the number of which is really
only determined by scientific taste. What is more, a
reference to the reflections above set forth respecting
the ideas of time, inertia, and the like, will surely con-
vince us that, accurately viewed, the entire fact has,
in all its aspects, not yet been perfectly apprehended.
The results The point of vicw reached has, as Newton expressly

re ached

havenoth- States, nothing to do with the " unknown causes " of
with the so- natural phenomena. That which in the mechanics of
"causes" the present day is called force is not a something that

of phenom-
ena, lies latent in the natural processes, but a measurable,

actual circumstance of motion, the product of the mass
into the acceleration. Also when we speak of the at-
tractions or repulsions of bodies, it is not necessary to
think of any hidden causes of the motions produced.
We signalise by the term attraction merely an actually
existing resemblance between events determined by con-
ditions of motion and the results of our volitional im-
pulses. In both cases either actual motion occurs or,
when the motion is counteracted by some other circum-
stance of motion, distortion, compression of bodies,
and so forth, are produced.

2. The work which devolved on genius here, was

THE PRINCIPLES OF DYNAMICS. 247

the noting of the connection of certain determinative The form of
elements of the mechanical processes. The precise es- chanicai
tablishment of the form of this connection was rather a f^the m^n
task for plodding research, which created the different icai origin.
concepts and principles of mechanics. We can de-
termine the true value and significance of these prin-
ciples and concepts only by the investigation of their
historical origin. In this it appears unmistakable at
times, that accidental circumstances have given to the
course of their development a peculiar direction, which
under other conditions might have been very different.
Of this an example shall be given.

Before Galileo assumed the familiar fact of the de- For exam-
pendence of the final velocity on the time, and put it to leo's laws

, , . , , , , ^ of falling

the test of experiment, he essayed, as we have already bodies
seen, a different hypothesis, and made the final velocity taken a dif-
proportional to the space described. He imagined, by a
course of fallacious reasoning, likewise already referred
to, that this assumption involved a self-contradiction.
His reasoning was, that twice any given distance of de-
scent must, by virtue of the double final velocity ac-
quired, necessarily be traversed in the same time as the
simple distance of descent. But since the first half is
necessarily traversed first, the remaining half will have
to be traversed instantaneously, that is in an interval
of time not measurable. Whence, it readily follows,
that the descent of bodies generally is instantaneous.

The fallacies involved in this reasoning are manifest. Galileo's
Galileo was, of course, not versed in mental integra- and?u
tions, and having at his command no adequate methods
for the solution of problems whose facts were in any
degree complicated, he could not but fall into mistakes
whenever such cases were presented. If we call s the
distance and / the tinie, the Galilean assumption reads

errors.

248 THE SCIENCE OF MECHANICS.

in the language of to-day dsjdt =zas^ from which fol-
lows s^^Ai"^, where a is a constant of experience and
A a constant of integration. This is an entirely different
conclusion from that drawn by Galileo. It does not
conform, it is true, to experience, and Galileo would
probably have taken exception to a result that, as a
condition of motion generally, made s different from 0
when / equalled 0. But in itself the assumption is by
no means j-^^-contradictory.
The suppo- Let us suppose that Kepler had put to himself the
Kepler had same questiou. Whereas Galileo always sought after
leo's re- the Very simplest solutions of things, and at once re-

86 Aire tl fiS

jected hypotheses that did not fit, Kepler's mode of pro-
cedure was entirely different. He did not quail before
the most complicated assumptions, but worked his way,
by the constant gradual modification of his original
hypothesis, successfully to his goal, as the history of
his discovery of the laws of planetary motion fully
shows. Most likely, Kepler, on finding the assumption
dsjdt =^as would not work, would have tried a num-
ber of others, and among them probably the correct one
ds/dt z=a\/s. But from this would have resulted an
essentially different course of development for the sci-
ence of dynamics.
In such a It is Only gradually and with great difficulty that

c&sc the

concept the concept of * * work " has attained its present position

" work "

might have of importance ; and in our judgment it is to the above-

been the

original mentioned trifling historical circumstance that the diffi-
inechanics. cultics and obstacles it had to encounter are to be as-
cribed. As the interdependence of the velocity and the
time was, as it chanced, first ascertained, it could not
be otherwise than that the relation v = gt should appear
as the original one, the equation s = gt^/2 2iS the next
immediate, and gs = v^/2 as a remoter inference. In-

THE PRINCIPLES OF DYNAMICS. 249

troducing the concepts mass {nC) and force (/), where
p = mgy we obtain, by multiplying the three equations
by »*, the expressions mv=:^fty »ij=//*/2, fs=:^
tnv^ ji — the fundamental equations of mechanics. Of
necessity, therefore, the concepts force and momentum
{m v) appear more primitive than the concepts work (/ j)
and vis viva (jnv^). It is not to be wondered at, accord-
ingly, that, wherever the idea of work made its appear-
ance, it was always sought to replace it by the histor-
ically older concepts. The entire dispute of the Leib-
nitzians and Cartesians, which was first composed in
a manner by D'Alembert, finds its complete explana-
tion in this fact.

From an unbiassed point of view, we have exactly Jastifica-

, . tion of this

the same right to inquire after the interdependence of view,
the final velocity and the time as after the interde-
pendence of the final velocity and the distance, and to
answer the question by experiment. The first inquiry
leads us to the experiential truth, that given bodies in
contraposition impart to each other in given times defi-
nite increments of velocity. The second informs us,
that given bodies in contraposition impart to each other
for given mutual displacements definite increments of
velocities. Both propositions are equally justified, and
both may be regarded as equally original.

The correctness of this view has been substantiated Exempiifi-

,, 1 A t ^ -r^ ^m -»«• cation of it

m our own day by the example of J. R. Mayer. Mayer, in modem
a modem mind of the Galilean stamp, a mind wholly
free from the influences of the schools, of his own in-
dependent accord actually pursued the last-named
method, and produced by it an extension of science
which the schools did not accomplish until later in a
much less complete and less simple form. For Mayer,
work was the original concept. That which is called

aso THE SCIENCE OF MECHANICS,

work in the mechanics of the schools, he calls force.

Mayer's error was, that he regarded his method as the

only correct one.
Thnxvmsttk 3. We may, therefore, as it suits us, regard the time
from it. of dcscent or the distance of descent as the factor de-

terminative of velocity. If ^ro fix our attention on
the first circumstance, the concept of force appears as
the original notion, the concept of work as the derived
one. If we investigate the influence of the second fact
first, the concept of work is the original notion. In
the transference of the ideas reached in the observation
of the motion of descent to more complicated relations,
force is recognised as dependent on the distance be-
tween the bodies — that is, as a function of the distance,
/(r). The work done through the element of distance dr
is then/(r) dr. By the second method of investiga-
tion work is also obtained as a function of the distance,
F (r) ; but in this case we know force only in the form
d. F{r)/dr — that is to say, as the limiting value of the
ratio : (increment of work)/(increment of distance.)
The prefer- Galileo cultivated by preference the first of these
different in- two methods. Newtou likewise preferred it. Huygens
pursued the second method, without at all restricting
himself to it. Descartes elaborated Galileo's ideas after
a fashion of his own. But his performances are in-
significant compared with those of Newton and Huy-
gens, and their influence was soon totally effaced. After
Huygens and Newton, the mingling of the two spheres
of thought, the independence and equivalence of which
are not always noticed, led to various blunders and
confusions, especially in the dispute between the Car-
tesians and Leibnitzians, already referred to, concern-
ing the measure of force. In recent times, however, in-
quirers turn by preference now to the one and now to

THE PRINCIPLES OF DTNAMaCS. . 251

the other. Thus the Galileo-Newtonian ideas are culti-
vated with preference by the school of Poinsot, the
Galileo- Huygenian by the school of Poncelet.

4. Newton operates almost exclusively with the no- The impor-

, , tance and

tions of force, mass, and momentum. His sense of the history of
value of the concept of m^ss places him above his prede- toman con-
cessors and contemporaries. It did not occur to Galileo mass.
that mass and weight were different things. Huygens,
too, in all his considerations, puts weights for masses ;
as for example in his investigations concerning the
centre of oscillation. Even in the treatise De Percus-
stone (On Impact), Huygens always says ''corpus ma-
jus," the larger body, and '* corpus minus," the smaller
body, when he means the larger or the smaller mass.
Physicists were not led to form the concept mass till
they made the discovery that the same body can by the
action of gravity receive different accelerations. The
first occasion of this discovery was the pendulum-ob-
servations of Richer (i 671-1673), — from which Huy-
gens at once drew the proper inferences, — and the
second was the extension of the dynamical laws to the
heavenly bodies. The importance of the first point may
be inferred from the fact that Newton, to prove the pro-
portionality of mass and weight on the same spot of the
earth, personally instituted accurate observations on
pendulums of different materials {Principia, Lib. II,
Sect. VI, De Motu et Resistentia Corporum Funependu-
lorutn). In the case of John Bernoulli, also, the first
distinction between mass and weight (in the Meditatio
de Natura Centri Oscillaiionis, Opera Omnia^ Lausanne
and Geneva, Vol. II, p. 168) was made on the ground
of the fact that the same body can receive different
gravitational accelerations. Newton, accordingly, dis-
poses of all dynamical questions involving the relations

252 . THE SCIENCE OF MECHANICS,

of several bodies to each other, by the help of the ideas
of force, mass, and momentum.
Tbemeth- 5. Huygens pursued a different method for the so-
gana. "^ lution of these problems. Galileo had previously dis-
covered that a body rises by virtue of the velocity ac-
quired in its descent to exactly the same height as that
from which it fell. Huygens, generalising the principle
(in his Horologium Oscillaiorium) to the effect that the
centre of gravity of any system of bodies will rise by
virtue of the velocities acquired in its descent to ex-
actly the same height as that from which it fell, reached
the principle of the equivalence of work and vis viva.
The names of the formulae which he obtained, were,
of course, not supplied until long afterwards.

The Huygenian principle of work was received by
the contemporary world with almost universal distrust.
People contented themselves with making use of its
brilliant consequences. It was always their endeavor
to replace its deductions by others. Even after John
and Daniel Bernoulli had extended the principle, it
was its fruitfulness rather than its evidency that was
valued.
The meth- We observe, that the Galileo-Newtonian principles
ton and Were, ou accouut of their greater simplicity and ap-
compared. pareutly greater evidency, invariably preferred to the
Galileo-Huygenian. The employment of the latter is
exacted only by necessity in cases in which the em-
ployment of the former, owing to the laborious atten-
tion to details demanded, is impossible ; as in the case
of John and Daniel Bernoulli's investigations of the
motion of fluids.

If we look at the matter closely, however, the same
simplicity and evidency will be found to belong to the
Huygenian principles as to the Newtonian proposi-

THE PRINCIPLES OF DYNAMICS. 253

tions. That the velocity of a body is determined by
the time of descent or determined by the distance of
descent, are assumptions equally natural and equally
simple. The form of the law must in both cases be
supplied by experience. As a starting-point, therefore,
pt =zmv and /J =zmv^ /2 are equally well fitted.

6. When we pass to the investigation of the motion The nece»-
of several bodies, we are again compelled, in both cases, unfversai-
to take a second step of an equal degree of certainty, two meth-
The Newtonian idea of mass is justified by the fact,
that, if relinquished, all rules of action for events would
have an end ; that we should forthwith have to expect
contradictions of our commonest and crudest experi-
ences ] and that the physiognomy of our mechanical
environment would become unintelligible. The same
thing must be said of the Huygenian principle of work.
If we surrender the theorem 2ps =z2mv^ /2j heavy
bodies will, by virtue of their own weights, be able to
ascend higher ; all known rules of mechanical occur-
rences will have an end. The instinctive factors which
entered alike into the discovery of the one view and of
the other have been already discussed.

The two spheres of ideas could, of course, have The points
grown up much more independently of each other. But of the two
in view of the fact that the two were constantly in con-
tact, it is no wonder that they have become partially
merged in each other, and that the Huygenian appears
the less complete. Newton is all-sufficient with his
forces, masses, and momenta. Huygens would like-
wise suffice with work, mass, and vis viva. But since
he did not in his time completely possess the idea of
mass, that idea had in subsequent applications to be
borrowed from the other sphere. Yet this also could
have been avoided. If with Newton the mass-ratio of

254 ^^^ SCIENCE OF MECHANICS,

two bodies can be defined as the inverse ratio of the
velocities generated by the same force, with Huygens
it would be logically and consistently definable as the
inverse ratio of the squares of the velocities generated
by the same work.
The re8i>ec- The two spheres of ideas consider the mutual de-
of each, pendence on each other of entirely different factors of
the same phenomenon. The Newtonian view is in so
far more complete as it gives us information regarding
the motion of each mass. But to do this it is obliged
to descend greatly into details. The Huygenian view
furnishes a rule for the whole system. It is only a con-
venience, but it is then a mighty convenience, when
the relative velocities of the masses are previously and
independently known.
Theeen- 7. Thus wc are led to see, that in the develop-

eral devel- ...

opment of ment of dynamics, just as in the development of statics,

dynamics

in the li^ht the Connection of widely diHerent features of mechanical

of the pre- j i » « • 1

ceding re- phenomena engrossed at different times the attention
of inquirers. We may regard the momentum of a sys-
teni as determined by the forces ; or, on the other
hand, we may regard its vis viva as determined by the
work. In the selection of the criteria in question the
individuality of the inquirers has great scope. It will
be conceived possible, from the arguments above pre-
sented, that our system of mechanical ideas might,
perhaps, have been different, had Kepler instituted
the first investigations concerning the motions of fall-
ing bodies, or had Galileo not committed an error in
his first speculations. We shall recognise also that not
only a knowledge of the ideas that have been accepted
and cultivated by subsequent teachers is necessary for
the historical understanding of a science, but also that
the rejected and transient thoughts of the inquirers.

THE PRINCIPLES OF D YNAMICS. 255

nay even apparently erroneous notions, may be very
important and very instructive. The historical investi-
gation of the development of a science is most needful,
lest the principles treasured up in it become a system
of half-understood prescripts, or worse, a system of
prejudices. Historical investigation not only promotes
the understanding of that which now is, but also brings
new possibilities before us, by showing that which ex-
ists to be in great measure conventional and accidental.
From the higher point of view at which different paths
of thought converge we may look about us with freer
powers of vision and discover routes before unknown.

In all the dynamical propositions that we have dis- The sabsti-

. - . , . iM mi tution of

cussed, velocity plays a promment role. The reason "intecrai**
of this, in our view, is, that, accurately considered, entiai"

laws maT

every single body of the universe stands in some defi- some day
nite relation with every other body in the universe ; concept of
that any one body, and consequently also any several fluous.
bodies, cannot be regarded as wholly isolated. Our
inability to take in all things at a glance alone compels
us to consider a few bodies and for the time being to
neglect in certain aspects the others ; a step accom-
plished by the introduction of velocity, and therefore
of time. We cannot regard it as impossible that inte-
gral laws, to use an expression of C. Neumann, will
some day take the place of the laws of mathematical
elements, or differential laws, that now make up the
science of mechanics, and that we shall have direct
knowledge of the dependence on one another of the
positions of bodies. In such an event, the concept of
force will have become superfluous.

CHAPTER III.

THE EXTENDED APPLICATION OF THE PRINCIPLES
OF MECHANICS AND THE DEDUCTIVE DE-
VELOPMENT OF THE SCIENCE.

SCOPE OF THE NEWTONIAN PRINCIPLES.

Newton's I. The principles of Newton suffice by themselves,

Src"um-** without the introduction of any new laws, to explore
scope and thoroughly every mechanical phenomenon practically
^^*^'' occurring, whether it belongs to statics or to dynamics.
If difficulties arise in any such consideration, they are

invariably of a mathematical, or
formal, character, and in no re-
spect concerned with questions
of principle. We have given,
let us suppose, a number of mas-
ses/n^, fn2t ^3- . - . in space, with
definite initial velocities »j, v^,
?;,.... We imagine, further, lines
of junction drawn between every
two masses. In the directions of
these lines of junction are set up the accelerations and
counter-accelerations, the dependence of which on the
distance it is the business of physics to determine. In
a small element of time r the mass m^, for example,
will traverse in the direction of its initial velocity the
distance v^t, and in the directions of the lines joining

THE EXTENSION OF THE PRINCIPLES. 257

it with the masses m^, m^, m^. . , ., being afiected in Schematic
such directions with the accelerations (p^, (p^, ^|. . . ., of the pre-

ceo 1 TW

the distances (^<p\/2)r^, (<^|/2)t2, (<^|/2)r2. . . . If statement.

we imagine all these motions to be performed indepen-
dently of each other, we shall obtain the new position
of the mass m^ after lapse of time r. The composition
of the velocities v^ and (p^r, (p%r, (p\r.,.. gives the
new initial velocity at the end of time t. We then
allow a second smalf interval of time r to elapse, and,
making allowance for the new spatial relations of the
masses, continue in the same way the investigation of
the motion. In like manner we may proceed with
every other mass. It will be seen, therefore, that, in
point of principle, no embarrassment can arise ; the
difficulties which occur are solely of a mathematical
character, where an exact solution in concise symbols,
and not a clear insight into the momentary workings
of the phenomenon, is demanded. If the accelerations
of the mass wig, or of several masses, collectively neu-
tralise each other, the mass m^ or the other masses
mentioned are in equilibrium and will move uniformly
onwards with their initial velocities. If, in addition,
the initial velocities in question are = 0, both equilib-
rium and rest subsist for these masses.

Nor, where a number of the masses m,, m^ , . , , The same

7 idea ap-

have considerable extension, so that it is impossible to pUed to aR-

Rrcgates of

speak of a single line joining every two masses, is the dif- material

, , particles.

ficulty, in point of principle, any greater. We divide
the masses into portions sufficiently small for our pur-
pose, and draw the lines of junction mentioned between
every two such portions. We, furthermore, take into
account the reciprocal relation of the parts of the
same large mass ; which relation, in the case of rigid
masses for instance, consists in the parts resisting

358 THE SCIENCE OF MECHANICS.

every alteration of their distances from one another.
On the alteration of the distance between any two .parts
of such a mass an acceleration is observed proportional
to that alteration. Increased distances diminish, and
diminished distances increase in consequence of this
acceleration. By the displacement of the parts with
respect to one another, the familiar forces of elasticity
are aroused. When masses meet in impact, their
forces of elasticity do not come into play until contact
and an incipient alteration of form take place.
A practical 2. If we imagine a heavy perpendicular column
of?he sco% resting on the earth, any particle m in the interior of
principles." the columu which we may choose to isolate in thought,
is in equilibrium and at rest. A vertical downward ac-
celeration g is produced by the earth in the particle,
which acceleration the particle obeys. But in so doing
it approaches nearer to the particles lying beneath it,
and the elastic forces thus awakened generate in x» a
vertical acceleration upwards, which ultimately, when
the particle has approached near enough, becomes
equal to g. The particles lying above m likewise
approach m with the acceleration g. Here, again,
acceleration and counter-acceleration are produced,
whereby the particles situated above are brought to
rest, but whereby m continues to be forced nearer and
nearer to the particles beneath it until the acceleration
downwards, which it receives from the particles above
it, increased by g, is equal to the acceleration it re-
ceives in the upward direction from the particles be-
neath it. We may apply the same reasoning to every
portion of the column and the earth beneath it, readily
perceiving that the lower portions lie nearer each other
and are more violently pressed together than the parts
above. Every portion lies between a less closely pressed

THE EXTENSION OF THE PRINCIPLES, 259

upper portion and a more closely pressed lower por- Rest in the
tion ; its downward acceleration g is neutralised by a these prin-
surplus of acceleration upwards, which it experiences pears as a

SDecial case

from the parts beneath. We comprehend the equilib- ofmotion.
rium and rest of the parts of the column by imagining
all the accelerated motions which the reciprocal rela-
tion of the earth and the parts of the column determine,
as in fact simultaneously performed. The 'apparent
mathematical sterility of this conception vanishes, and
it assumes at once an animate form, when we reflect
that in reality no body is completely at rest, but that
in all, slight tremors and disturbances are constantly
taking. place which now give to the accelerations of de-
scent and now to the accelerations of elasticity a slight
preponderance. Rest, therefore, is a case of motion,
very infrequent, and, indeed, never completely realised.
The tremors mentioned are by no means an unfamiliar
phenomenon. When, however, we occupy ourselves
with cases of equilibrium, we are concerned simply with
a schematic reproduction in thought of the mechanical
facts. We then purposely neglect these disturbances,
displacements, bendings, and tremors, as here they
have no interest for us. All cases of this class, which
have a scientific or practical importance, fall within the
province of the so-called theory of elasticity. The whole The unity

, and homo-

outcome of Newton's achievements is that we every- geneity

which tliAftA

where reach our goal with one and the same idea, and principles

1 1" • 1 1 t 1 \ introduce

by means of it are able to reproduce and construct be- into the

science

forehand all cases of equilibrium and motion. All
phenomena of a mechanical kind now appear to us
as uniform throughout and as made up of the same
elements.

3. Let us consider another example. Two mas-
ses M, m are situated at a distance a from each

26o

THE SCIENCE OF MECHANICS,

A general
exemplifi-
cation of
the power
of the prin-
ciples.

The devel-
opment of
the equa-
tions ob-
tained in
this exam-
ple.

S=&—

Other. (Fig. 145.) When displaced with respect to
each other, elastic forces proportional to the change
x^ of distance are supposed to be

awakened. Let the masses be
movable in the A'-direction par-
Fig- 145. allel to tf, and their codrdinates
be jCj, ^Cj. If a force /is applied at the point x^^ the
following Equations obtain :

d^x.

(1)
(2)

where / stands for the force that one mass exerts on
the other when their mutual distance is altered by the
value I. All the quantitative properties of the me-
chanical process are determined by these equations.
But we obtain these properties in a more comprehensi-
ble form by the integration of the equations. The ordi-
nary procedure is, to find by the repeated differentia-
tion of the equations before us new equations in suffi-
cient number to obtain by elimination equations in x^
alone or x^ alone, which are afterwards integrated. We
shall here pursue a different method. By subtracting
the first equation from the second, we get

d^{x^ — x,^_
dt^

— 2/[(^2 — ^1) — «]+/» or

putting x^ — x^ = Uy

d^U „ ^ -I . y

(3)

and by the addition of the first and the second equa-
tions

d^ (x^ + x^) .
m -^- — ^ =/, or, puttmg x^ + x^= v,

di^

THE EXTENSION OF THE PRINCIPLES, 261

^77^=^ ••• W

The integrals of (3) and (4) are respectively The integ-

rals of these

u = A sin-^r~ . t -{- B cos-vl— • ^ + <3f + ^ and
v= -- , -^ -\- Ci -\- D\ whence

developH
ments.

+ '-' 2 4/+ 2'

*» = -2 «"'>| ^,- • ' + 2 *=°^\ «r • ' + L • 2-

+ ^'+2+47+"^-
To take a particular case, we will assume that the a partica-

- , - - , . ^ - , , . lar case of

action 01 the force/ begins at /= 0, and that at this the exam-
ple,
time

.. = 0.^ = 0

x^ = tf, —j^ = 0,

that is, the initial positions are given and the initial
velocities are = 0. The constants Ay B, C, D being
eliminated by these conditions, we get

(5) -x=47'^os^lf-'+2^-Y-47'
(7) ^.-*x=-2^cos^^^.'+« + 2>-

262 THE SCIENCE OF MECHANICS.

Theinfonn- We scc from (5) and (6) that the two masses, in addi-

ation which . \jj \ j »

the result- tion to a Uniformly accelerated motion with half the

ant eQua-

tions give acceleration that the force / would impart to one of

concerning

this exam- these masses alone, execute an oscillatory motion sym-
metrical with respect to their centre of gravity. The
duration of this oscillatory motion, 7'= 7.nVnil7.p^ is
smaller in proportion as the force that is awakened in
the same mass-displacement is greater (if our attention
is directed to two particles of the same body, in. pro-
portion as the body is harder). The amplitude of os-
cillation of the oscillatory motion fjip likewise de-
creases with the magnitude / of the force of displace-
ment generated. Equation (7) exhibits the periodic
change of distance of the two masses during their pro-
gressive motion. The motion of an elastic body might
in such case be characterised as vermicular. With hard
bodies, however, the number of the oscillations is so
great and their excursions so small that they remain
unnoticed, and may be left out of account. The oscil-
latory motion, furthermore, vanishes, either gradually
through the effect of some resistance, or when the two
masses, at the moment the force /begins to act, are a
distance a -\-f/2p apart and have equal initial veloci-
ties. The distance a -\- ffip that the masses are apart
after the vanishing of their vibratory motion, is//2/
greater than the distance of equilibrium a. A tension
yy namely, is set up by the action of/, by which the
acceleration of the foremost mass is reduced to one-
half whilst that of the mass following is increased by
the same amount. In this, then, agreeably to our as-
Thia in- sumption, pyjni ■=^fl7. m ox y =//2/. As we see, it is
u exhaus- in our power to determine the minutest details of a

tive*

phenomenon of this character by the Newtonian prin-
ciples. The investigation becomes (mathematically,

THE EXTENSION OF THE PRINCIPLES. 263

yet not in point of principle) more complicated when
we conceive a body divided up into a great number of
small parts that cohere by elasticity. Here also in the
case of sufficient hardness the vibrations may be neg-
lected. Bodies in which we pur|X)sely regard the mu-
tual displacement of the parts as evanescent, are called
rigid bodies.

4. We will now consider a case that exhibits theihededuc-

. . tion of the

schema of a lever. We imagine the masses M^ m^, ni^ laws of the
arranged in a triangle and joined by elastic connec- Newton's
tions. Every alteration of the sides, and consequently
also every alteration of the angles, gives rise to accel-
erations, as the result of which the triangle endeavors to
assume its previous form and size. By the aid of the
Newtonian principles we can deduce from such a
schema the laws of the lever, and at the same time feel
that the form of the deduction, although it may be
more complicated, still
remains admissible when
we pass from a schematic
lever composed of three
masses to the case of a
real lever. The mass M
we assume either to be in itself very large or conceive
it joined by powerful elastic forces to other very large
masses (the earth for instance). Af then represents
an immovable fulcrum.

Let m^, now, receive from the action of some ex-Themeth-
temal force an acceleration /perpendicular to the line deduction,
of junction Mm^ = c -\- d. Immediately a stretching
of the lines m^m^ = ^ and m^M^a is produced, and
in the directions in question there are respectively set
up the accelerations, as yet undetermined, s and <T, of
which the components s(js/b) and (J^eja^ are directed

Fig. 146.

264 ^^^^ SCIENCE OF MECHANICS,

Oppositely to the acceleration/. Here e is the altitude
of the triangle m^m^M, The mass m^ receives the
acceleration s\ which resolves itself into the two com-
ponents j'(////^) in the direction of-Afand s\e/b)'gdLT'
allel to f. The former of these determines a slight ap-
proach of m^ to M, The accelerations produced in M
by the reactions ol m^ and m^, owing to its great mass,
are imperceptible. We purposely neglect, therefore,
the motion of M,
The deduc- The mass m^y accordingly, receives the accelera-
taincdby tion / — s(e/b) — a{e/a), whilst the mass m^ suffers
eration of the parallel acceleration s\e/b). Between s and a a
tions. simple relation obtains. If, by supposition, we have a
very rigid connection, the triangle is only impercept-
ibly distorted. The components of s and ff perpendicular
to / destroy each other. For if this were at any one
moment not the case, the greater component would
produce a further distortion, which would immediately
counteract its excess. The resultant of s and <T is
therefore directly contrary to/, and consequently, as is
readily obvious, a {^/d) = s {d/3). Between s and s\
further, subsists the familiar relation m^s ^^m^s' or
s = s'{m^/m^). Altogether m^ and m^ receive re-
spectively the accelerations s\e/lf) and / — s\e/^^
(jn^/m^){c -\- d/e), or, introducing in the place of the
variable value s\e/d) the designation <p, the accelera-
tions q} and/ — ^(^2/^1) (^ "i" ^A)-
On the pre- At the Commencement of the distortion, the accel-
positioDs eration oi m^y owing to the increase of (p, diminishes,
the rotation whilst that of m2 iucreases. If we make the altitude e
are easily of the triangle very small, our reasoning still remains
applicable. In this case, however, a becomes -— e = r^,
and a-^^ = c-\-d=r2. We see, moreover, that the
distortion must continue, ^ increase, and the accelera-

THE EXTENSION OF THE PRINCIPLES, 265

tion of Wj diminish until the stage is reached at which
the accelerations oim^ and tn^ bear to each other the
proportion of r^ to rg. This is equivalent to a rotation
of the whole triangle (without further distortion) about
M^ which mass by reason of the vanishing accelera-
tions is at rest. As soon as rotation sets in, the rea-
son for further alterations of <p ceases. In such a case,
consequently,

9 =

'*2

^2 '*2l ^^ ^ ^ ^l^\f

/— . (p -2 -2^ or (p =

r^ -

;;/^ r^ J ' ^ ^ m^ ''1^ + ^^\ *" ^

2 ' 2
For the angular acceleration ^ of the lever we get

^2 '^1^1^ + ^2^2^
Nothing prevents us from entering still more into Discussion
the details of this case and determining the distortions acter of the

" preceding

and vibrations of the parts with respect to each other, result.
With sufficiently rigid connections, however, these de-
tails may be neglected. It will be perceived that we .
have arrived, by the employment of the Newtonian prin-
ciples, at the same result to which the Huygenian view
also would have led us. This will not appear strange to
us if we bear in mind that the two views are in every re-
spect equivalent y and merely start from different aspects
of the same subject-matter. If we had pursued the
Huygenian method, we should have arrived more
speedily at our goal but with less insight into the de-
tails of the phenomenon. We should have employed
the work done in some displacement of /w^ to deter-
mine the vires viva oi tn^ and m^^ wherein we should
have assumed that the velocities in question v^^ v^
maintained the ratio v^lv^^=r^lr^. The example
here treated is very well adapted to illustrate what
such an equation of condition means. The equation

266

THE SCIENCE OF MECHANICS.

A simple
case of the
same exam
pie.

The equi-
librium of
the lever
deduced
from the
same con-
siderationa.

simply asserts, that on the slightest deviations of v^jv^
from r^jr^ powerful forces are set in action which in
point of fact prevent all further deviation. The bodies
obey of course, not the equations^ but \\\<& forces.

5. We obtain a very obvious case if we put in the
example just treated m^^=^m^=^m and a = ^(Fig.
147). The dynamical state of the system ceases to
change when ^ = 2 (/ — 2 ^), that is, when the accel-
erations of the masses
at the base and the ver-
ak^'^ /i^^^^^i/ tex are given by 2//5
wT . and //5. At the com-

^'^' '"'• mencement of the dis-

tortion <p increases, and simultaneously the accelera-
tion of the mass at the vertex is decreased by double
that amount, until the proportion subsists between the
two of 2 : 1.

We have yet to consider the case of equilibrium of
a schematic lever, consisting (Fig. 148) of three masses
m^, m^, and M, of which the last is again supposed

Fig. 148.

to be very large or to be elastically connected with
very large masses. We imagine two equal and oppo-
site forces s, — s applied to m^ and m^ in the direction
m^m^, or, what is the, same thing, accelerations im-
pressed inversely proportional to the masses m^^, m^.
The stretching of the connection m^m^ also generates

THE EXTENSION OF THE PRINCIPLES. 267

accelerations inversely proportional to the masses m^^
tn^y which neutralise the first ones and produce equi-
librium. Similarly, along m^M imagine the equal and
contrary forces /, — / operative ; and along m^M the
forces Uy — u. In this case also equilibrium obtains.
If M be elastically connected with masses sufficiently
large, — u and — / need not be applied, inasmuch
as the last-named forces are spontaneously evoked the
moment the distortion begins, and always balance the
forces opposed to them. Equilibrium subsists, accord-
ingly, for the two equal and opposite forces x, — x as
well as for the wholly arbitrary forces /, u, Aaa matter
of fact X, — s destroy each other and /, u pass through
the fixed mass My that is, are destroyed on distortion
setting in.

The condition of equilibrium readily reduces itself The reduo-
to the common form when we reflect that the mo- preceding
ments of / and », forces passmg through My are with common
respect to M zero, while the moments of s and — s are
equal and opposite. If we compound / and s to /, and
u and — X to ^, then, by Varignon's geometrical principle
of the parallelogram, the moment of / is equal to the
sum of the moments of s and /, and the moment of q
is equal to the sum of the moments of u and — x. The
moments of/ and q are therefore equal and opposite.
Consequently, any two forces / and q will be in equi-
librium if they produce in the direction m^m^ equal
and opposite components, by which condition the equal-
ity of the moments with respect to M is posited. That
then the resultant of / and q also passes through My is
likewise obvious, for x and — x destroy each other and
/ and u pass through M,

6. The Newtonian point of view, as the example
just developed shows us, includes that of Varignon.

268 THE SCIENCE OF MECHANICS,

Newton's We Were right, therefore, when we characterised the

point of . .

view in- statics of Varignon as a dynamical statics, which, start-
varignon's. ing from the fundamental ideas of modern dynamics,
voluntarily restricts itself to the investigation of cases
of equilibrium. Only in the statics of Varignon, owing
to its abstract form, the significance of many opera-
tions, as for example that of the translation of the
forces in their own directions, is not so distinctly ex-
hibited as in the instance just treated.
The econ- The Considerations here developed will convince
wealth of US that we can dispose by the Newtonian principles

the N evirtOQ-

ian ideas, of every phenomenon of a mechanical kind which may
arise, provided we only take the pains to enter far
enough into details. We literally see through the cases
of equilibrium and motion which here occur, and be-
hold the masses actually impressed with the accelera-
tions they determine in one another. It is the same
grand fact, which we recognise in the most various
phenomena, or at least can recognise there if we make
a point of so doing. Thus a unity, homogeneity, and
economy of thought were produced, and a new and
wide domain of physical conception opened which
before Newton's time was unattainable.
The New- Mechauics, however, is not altogether an end in it-

the modem, self ; it has ^Xs^o. problems to solve that touch the needs
methods, of practical life and affect the furtherance of other sci-
ences. Those problems are now for the most part ad-
vantageously solved by other methods than the New-
tonian,— methods whose equivalence to that has already
been demonstrated. It would, therefore, be mere im-
practical pedantry to contemn all other advantages and
insist upon always going back to the elementary New-
tonian ideas. It is sufficient to have once convinced
ourselves that this is always possible. Yet the New-

THE EXTENSION OF THE PRINCIPLES. 269

tonian conceptions are certainly the most satisfactory
and the most lucid ; and Poinsot shows a noble sense
of scientific clearness and simplicity in making these
conceptions the sole foundation of the science.

II.

THE FORMULiE AND UNITS OF MECHANICS.

1. All the important formulae of modern mechanics History of

• % e r^ \'% "** fonnu-

were discovered and employed in the period of Galileo las and

. . . . units of

and Newton. The particular designations, which, mechanics,
owing to the frequency of their use, it was found con-
venient to give them, were for the most part not fixed
upon until long afterwards. The systematical mechan-
ical units were not introduced until later still. Indeed,
the last named improvement, cannot be regarded as
having yet reached its completion.

2. Let 5 denote the distance, / the time, v the in- The orig-
stantaneous velocity, and €p the acceleration of a uni- tfons??*
formly accelerated motion. From the researches ofHuygens.
Galileo and Huygens, we derive the following equa-
tions :

V = <pt

(1)

Multiplying throughout by the mass m, these equa- The intro-

,1 r 11 • duction

tions give the following : of "mass

and "mov-
m%) = m(pt ^ force."

mm .

mcps -=1

mv^

2 '

270

THE SCIENCE OF MECHANICS

Final form and, denoting the moving force mq>\iy the letter /, we

of the fun-
dament&I Obtain

equations.

mv •= pt

• 2

2,"

ps =

(2)

Equations (i) all contain the quantity (p ; and each
contains in addition two of the quantities s^ /, Vy as
exhibited in the following table :

9 \

Equations (2) contain the quantities m^ /, Sy /, v ;
each containing m^ p and in addition to m^ p two of the
three quantities x, /, v^ according to the following table :

m,p

. S, V

The scope
and appli-

Questions concerning motions due to constant forces
catiotTof' are answered by equations (2) in great variety. If, for
tions. ***"" example, we want to know the velocity v that a mass
m acquires in the time / through the action of a force
/, the first equation gives v z=pt/m. If, on the other
hand, the time be sought during which a mass m with
the velocity v can move in opposition to a force /, the
same equation gives us / = /« v/p. Again, if we in-
quire after the distance through which m will move with
velocity v in opposition to the force /, the third equa-
tion gives s = tnv^ jip. The two last questions illus-
trate, also, the futility of the Descartes- Leibnitzian dis-
pute concerning the measure of force of a body in mo-
tion. The use of these equations greatly contributes

THE EXTENSION OF THE PRINCIPLES. 271

to confidence in dealing with mechanical- ideas. Sup-
pose, for instance, we put to ourselves, the question,
what force p will impart to a given mass m the velocity
V \ we readily see that between »i, /, and v alone, no
equation exists, so that either x or / must be supplied,
and consequently the question is an indeterminate one.
We soon learn to recognise and avoid indeterminate
cases of this kind. The distance that a mass m acted
on by the force / describes in the time /, if moving
with the initial velocity 0, is found by the second equa-
tion s=ipt^/2m.

3. Several of the formulae in the above-discussed The names
equations have received particular names. The force formula of
of a moving body was spoken of by Galileo, who al- tionsHave
ternately calls it ''momentum," "impulse," and "en-
ergy. " He regards this momentum as proportional to
the product of the mass (or rather the weight, for Gali-
leo had no clear idea of mass^ and for that matter no
more had Descartes, nor even Leibnitz) into the velo-
city of the body. Descartes accepted this view. He put
the force of a moving body = w z/, called it quantity of
motion, and maintained that the sum-total of the quan-
tity of motion in the universe remained constant, so that
when one body lost momentum the loss was compen-
sated for by an increase of momentum in other bodies.
Newton also employed the designation "quantity of
motion " for m v, and this name has been retained to the Momen-

, _T-» . t , -, tum and

present day. [But momentum is the more usual term. J impulse.
For the second member of the first equation, viz. / /,
Belanger, proposed, as late as 1847, the name impulse*
The expressions of the second equation have received

* See, also, Maxwell, Matter and Motion^ American edition, page 7a. But
this word is commonly used in a different sense, namely, as "the limit of a
force which is infinitely great but acts only during an infinitely short time."
See Routh, Rigid Dynamic*^ Part I, pages 65-66.— TVaiw.

272 THE SCIENCE OF MECHANICS.

visvrva no particular designations. Leibnitz (1695) called the
expression mv'^ of the third equation vis viva or living
force, and he regarded it, in opposition to Descartes,
as the true measure of the force of a body in motion,
calling the pressure of a body at rest vis mortua, or
dead force. Coriolis found it more appropriate to give
the term \mv^ the name vis viva. To avoid confusion,
Belanger proposed to call mv^ Urging force and \mi^^
living power [now commonly called in English kinetic
energy]. For ps Coriolis employed the name work.
Poncelet confirmed this usage, and adopted the kilo-
gramme-metre (that is, a force equal to the weight of a
kilogramme acting through the distance of a metre) as
the unit of work.

The history a Concerning the historical details of the origin of

of the ideas ~ ° . . °

quantity of thcse notious "quantity of motion*' and "vis viva,"

motion and

vis viva, a glance may now be cast at the ideas which led Des-
cartes and Leibnitz to their opinions. In his Principia
Fhilosophice, published in 1644, II, 36, Descartes ex-
pressed himself as follows :

"Now that the nature of motion has been examined,
"we must consider its cause, which may be conceived
"in two senses : first, as a universal, original cause —
" the general cause of all the motion in the world ; and
"second, as a special cause, from which the individual
" parts of matter receive motion which before they did
"not have. As to the universal cause, it can mani-
"festly be none other than God, who in the beginning
" created matt-er with its motion and rest, and who now
" preserves, by his simple ordinary concurrence, on the
"whole, the same amount of motion and rest as he
"originally created. For though motion is only a con-
"dition of moving matter, there yet exists in matter
"a definite quantity of it, which in the world at large.

THE EXTENSION OF THE PRINCIPLES, 273 I

^< never increases or diminishes, although in single por- Passage

''tions it changes; namely, in this way, that we must cartes's . \

'* assume, in the case of the motion of a piece of matter

'* which is moving twice as fast as another piece, but in

*' quantity is only one half of it, that there is the same

** amount of motion in both, and that in the proportion

** as the motion of one part grows less, in the same pro-

** portion must the motion of another, equally large

'*part grow greater. We recognise it, moreover, as

**a perfection of God, that He is not only in Himself

'^ unchangeable, but that also his modes of operation

" are most rigorous and constant ; so that, with the ex-

"ception of the changes which indubitable experience

'*or divine revelation offer, and which happen, as our

** faith or judgment show, without any change in the

"Creator, we are not permitted to assume any others

"in his works — lest inconstancy be in anyway pre-

"dicated of Him. Therefore, it is wholly rational to

"assume that God, since in the creation of matter he

" imparted different motions to its parts, and preserves

"all matter in the same way and conditions in which

"he created it, so he ^wcvXKxXy preserves in it the same

* ' quantity of motion. *'

The merit of having first sought after a more uni- The merits

. and defects

versal and more fruitful point of view in mechanics, o' t)escar-

' . , tes's phys-

cannot be denied Descartes. This is the peculiar task »cai inquir-
ies.

of the philosopher, and it is an activity which con-
stantly exerts a fruitful and stimulating influence on
physical science.

Descartes, however, was infected with all the usual
errors of the philosopher. He places absolute confi-
dence in his own ideas. He never troubles himself to
put them to experiential test. On the contrary, a min-
imum of experience always suffices him for a maximum

274 THE SCIENCE OF MECHANICS.

of inference. Added to this, is the indistinctness of
« his conceptions. Descartes did not possess a clear

idea of mass. It is hardly allowable to say that Des-
cartes defined m v as momentum, although Descartes's
scientific successors, feeling the need of more definite
notions, adopted this conception. Descartes's greatest
error, however, — and the one that vitiates all his phys-
ical inquiries, — is this, that many propositions appear
to him self-evident a priori concerning the truth of
which experience alone can decide. Thus, in the two
paragraphs following that cited above (§§37-39) it is
asserted as a self-evident proposition that a body pre-
serves unchanged its velocity and direction. The ex-
periences cited in §38 should have been employed, not
as a confirmation of an h priori law of inertia, but as a
foundation on which this law in an empirical sense
should be based.
Leibnitz Dcscartes's view was attacked by Leibnitz (1686)

on quantity "^ ^ ^

of motion, in the Acta Eruditorumy in a little treatise bearing the
title : "A short Demonstration of a Remarkable Error
of Descartes and Others, Concerning the Natural Law
by which they think that the Creator always preserves
the same Quantity of Motion ; by which, however, the
Science of Mechanics is totally perverted. "

In machines in equilibrium, Leibnitz remarks, the
loads are inversely proportional to the velocities of dis-
placement ; and in this way the idea arose that the
product of a body (" corpus," " moles ") into its velocity
is the measure of force. This product Descartes re-
garded as a constant quantity. Leibnitz's opinion,
however, is, that this measure of force is only acci-
dentally the correct measure, in the case of the ma-
chines. The true measure of force is different, and
must be determined by the method which Galileo and

THE EXTENSION OF THE PRINCIPLES. 275

Huygens pursued. Every body rises by virtue of the Leibnitz on
velocity acquired in its descent to a height exactly ure of force,
equal to that from which it fell. If, therefore, we as-
sume, that the same "force" is requisite to raise a
body m a height 4^ as to raise a body ^m a height //,
we must, since we know that in the first case the ve-
locity acquired in descent is but twice as great as in
the second, regard the product of a "body" into the
square of its velocity as the measure of force.

In a subsequent treatise (1695), Leibnitz reverts to
this subject. He here makes a distinction between
simple pressure {vis mortua) and the force of a moving
body {vis viva), which latter is made up of the sum of
the pressure- impulses. These impulses produce, in-
deed, an "impetus" {mv), but the impetus produced
is not the true measure of force ; this, since the cause
must be equivalent to the effect, is (in conformity with
the preceding considerations) determined by mv^,
Leibnitz remarks further that the possibility of per-
petual motion is excluded only by the acceptance of his
measure of force.

Leibnitz, no more than Descartes, possessed a gen- The idea of

mass in

uine concept of mass. Where the necessity of such Leibnitz's

view

an idea occurs, he speaks of a body {corpus)^ of a load
{moles), of different-sized bodies of the same specific
gravity, and so forth. Only in the second treatise, and
there only once, does the expression "massa" occur,
in all probability borrowed from Newton. Still, to de-
rive any definite results from Leibnitz's theory, we must
associate with his expressions the notion of mass, as
his successors actually did. As to the rest, Leibnitz's
procedure is much more in accordance with the meth-
ods of science than Descartes's. Two things, however,
are confounded : the question of the measure of force

276 THE SCIENCE OF MECHANICS.

In a sense, and the question of the constancy of the sums 2mv and
andLeib- 2mv^, The two have in reality nothing to do with
each right, each Other. With regard to the first question, we now
know that both the Cartesian and the Leibnitzian meas-
ure of force, or, rather, the measure of the effective-
ness of a body in motion, have, each in a different
sense, their justification. Neither measure, however,
as Leibnitz himself correctly remarked, is to be con-
founded with the common, Newtonian, measure of
force.
The dis- With regard to the second question, the later in-

8ui?of mf^ vestigations of Newton really proved that iox free ma-
sundings. terial systems not acted on by external forces the Car-
tesian sum 2mz; is a constant ; and the investigations
of Huygens showed that also the sum 2mv^ is a con-
stant, provided work performed by forces does not alter
it. The dispute raised by Leibnitz rested, therefore,
on various misunderstandings. It lasted fifty-seven
years, till the appearance of D'Alembert's Traite de
dynamiquCy in 1743. To the theological ideas of Des-
cartes and Leibnitz, we shall revert in another place.
Theappii- 5- The three equations above discussed, though
the funda- they are only applicable to rectilinear motions produced
equations by Constant forces, may yet be considered the funda-
forces? * mental equations of mechanics. If the motion be recti-
linear but the force variable, these equations pass by a
slight, almost self-evident, modification into others,
which we shall here only briefly indicate, since mathe-
matical developments in the present treatise are wholly
subsidiary.

From the first equation we get for variable forces

mv = Ipdt '\- C, where / is the variable force, dt the

time-element of the action, { pdt the sum of all the

THE EXTENSION OF THE PRINCIPLES, 277

•

products / . dt from the beginning to the end of the
action, and C a constant quantity denoting the value
of m V before the force begins to act.

The second equation passes in like manner into the

form s =i\dt\^-dt + C^ + I>, with two so-called

constants of integration.

The third equation must be replaced by

-;-=Sf'"+c.

Curvilinear motion may always be conceived as the
product of the simultaneous combination of three rec-
tilinear motions, best taken in three mutually perpen-
dicular directions. Also for the components of the mo-
tion of this very general case, the above-given equa-
tions retain their significance.

6. The mathematical processes of addition, sub- The units of
traction, and equating possess intelligible meaning only
when applied to quantities of the same kind. We can-
not add or equate masses and times, or masses and
velocities, but only masses and masses, and so on.
When, therefore, we have a mechanical equation, the
question immediately presents itself whether the mem-
bers of the equation are quantities of f/ie same kind,
that is, whether they can be measured by the same unit,
or whether, as we usually say, the equation is homo-
geneous. The units of the quantities of mechanics will
form, therefore, the next subject of our investigations.

The choice of units, which are, as we know, quan-
tities of the same kind as those they serve to measure,
is in many cases arbitrary. Thus, an arbitrary mass is
employed as the unit of length, an arbitrary time as the
unit of time. The mass and the length employed as
units can be preserved ; the time can be reproduced

force.

278 THE SCIENCE OF MECHANICS.

■

Arbitrary by pendulum-experiments and astronomical observa-

units, and . _.. •••i •/•!• • r

derived or tions. But units like a unit of velocity, or a unit of
units. acceleration, cannot be preserved, and are much more
difficult to reproduce. These quantities are conse-
quently so connected with the arbitrary fundamental
units, mass, length, and time, that they can be easily
and at once derived from them. Units of this class
are called derived or absolute units. This latter desig-
nation is due to Gauss, who first derived the magnetic
units from the mechanical, and thus created the possi-
bility of a universal comparison of magnetic measure-
ments. The name, therefore, is of historical origin.
The de- As unit of velocity we might choose the velocity

of velocity, with which, Say, q units of length are travelled over in
tion. and unit of time. But if we did this, we could not express
the relation between the time /, the distance i*, and the
velocity v by the usual simple formula s=:zvt, but
should have to substitute for \t s = q,iJt, If, however,
we define the unit of velocity as the velocity with
which the unit of length is travelled over in unit of
time, we may retain the form s=:vL Among the de-
rived units the simplest possible relations are made
to obtain. Thus, as the unit of area and the unit of vol-
ume, the square and cube of the unit of length are al-
ways employed.

According to this, we assume then, that by unit ve-
locity unit length is described in unit time, that by unit
acceleration unit velocity is gained in unit time, that
by unit force unit acceleration is imparted to unit mass,
and so on.

The derived units depend on the arbitrary funda-
mental units ; they are functions of them. The func-
tion which corresponds to a given derived unit is called
its dimensions. The theory of dimensions was laid down

•

THE EXTENSION OF THE PRINCIPLES. 279

by Fourier, in 1822, in his Theory of Heat, Thus, if /The theory
denote a length, / a time, and m a mass, the dimen- sions.
sions of a velocity, for instance, are /// or //~i. After
this explanation, the following table will be readily un-
derstood :

NAMS8 SYMBOLS bIMBMSIOMS

Velocity v lt~^

Acceleration <p lt~^

Force / mlt-^

Momentum mv mit~^

Impulse // mli"^

Work ps tnl^t-'^

TT* • mii^ ,« . «

Vis viva -^— w/*/-a

Moment of inertia 0 ml^

Statical moment D ml^t-^

This table shows at once that the above-discussed equa-
tions are homogeneous, that is, contain only members of
the same kind. Every new expression in mechanics
might be investigated in the same manner.

7. The knowledge of the dimensions of a quantity The usefui-
is also important for another reason. Namely, if the theoir of ^
value of a quantity is known for one set of fundamental s/oSIf
units and we wish to pass to another set, the value of
the quantity in the new units can be easily found from
the dimensions. The dimensions of an acceleration,
which has, say, the numerical value 9?, are //~2. if
we pass to a unit of length X times greater and to a
unit of time t times greater, then a number X times
smaller must take the place of / in the expression //"*,
and a number t times smaller the place of /. The
numerical value of the same acceleration referred to
the new units will consequently be (t* A) <p. If we

28o

THE SCIENCE OF MECHANICS.

The Inter-
national
Bureau of
Weights
and Meas-
ures.

The inter-
national
unit of
length.

take the metre as our unit of length, and the second as
our unit of time, the acceleration of a falling body for
example is 9-81, or as it is customary to write it, in-
dicating at once the dimensions and the fundamental
measures: 9-81 (metre/second 2). If we pass now to
the kilometre as our unit of length (A, = 1000), and to
the minute as our unit of time (t = 60), the value of the
same acceleration of descent is (60 X 60/1000)9-81,
or 35-316 (kilometre/minute 2).

[8. The following statement of the mechanical units
at present in use in the United States and Great Britain
is substituted for the statement by Professor Mach of
the units formerly in use on the continent of Europe.
All the civilised governments have united in establish-
ing an International Bureau of Weights and Measures
in the Pavilion de Breteuil, in the Pare of St. Cloud,
at Sevres, near Paris. In some countries, the stan-
dards emanating from this office are exclusively legal ;
in others, as the United States and Great Britain, they
are optional in contracts, and are usual with physi-
cists. These standards are a standard of length and a
standard of mass (not weight.^

The unit of length is the International Metre, which
is defined as the distance at the melting point of ice
between the centres of two lines engraved upon the
polished surface of a platiniridium bar, of a nearly
X-shaped section, called the International Prototype
Metre. Copies of this, called National Prototype Me-
tres, are distributed to the different governments. The
international metre is authoritatively declared to be
identical with the former French metre, used until the
adoption of the international standard ; and it is im-
possible to ascertain any error in this statement, be-

THE EXTENSION OF THE PRINCIPLES, 281

cause of doubt as to the length of the old metre,
owing partly to the imperfections of the standard, and
partly to obstacles now intentionally put in the way of
such ascertainment. The French . metre was defined
as the distance, at the melting-point of ice, between
the ends of a platinum bar, called the metre des archives.
It was against the law to touch the ends, which made
it difficult to ascertain the distance between them.
Nevertheless, there was a strong suspicion they had
been dented. The metre des archives was intended to
be one ten-millionth of a quadrant of a terrestrial
meridian. In point of fact such a quadrant is, ac-
cording to Clarke, 32814820 feet, which is 100020 15
metres.

The international unit of mass is the kilogramme. The inter-
national
which is the mass of a certain cylinder of platiniridium unit of

mass.

called the International Prototype Kilogramme. Each
government has copies of it called National Prototype
Kilogrammes. This mass was intended to be identical
with the former French kilogramme, which was defined
as the mass of a certain platinum cylinder called the
kilogramme des archives. The platinum being somewhat
spongy contained a variable amount of occluded gases,
ajid had perhaps suffered some abrasion. The kilo-
gramme is 1000 grammes ; and a gramme was intended
to be the mass of a cubic centimetre of water at its
temperature of maximum density, about 3 • 93° C. It
is not known with a high degree of precision how nearly
this is so, owing to the difficulty of the determination.

The regular British unit of length is the Imperial The British
Yard which is the distance at 62° F. between the cen- iSJgth.
tres of two lines engraved on gold plugs inserted in a
bronze bar usually kept walled up in the Houses of
Parliament in Westminster. These lines are cut rela-

282 THE SCIENCE OF MECHANICS.

Conditions tivelv deep, and the burr is rubbed off and the surface

of compari-
son of the rendered mat, by rubbing with charcoal. The centre

Imperial •

Yard with of such a line can easily be displaced by rubbing : which

other meas- . , , , r « « • i -r-^

urea. is probably not true of the lines on the Prototype me-

tres. The temperature is, by law, ascertained by a
mercurial thermometer ; but it was not known, at the
time of the construction of the standard, that such
thermometers may give quite different readings, ac-
cording to the mode of their manufacture. The quality
of glass makes considerable difference, and the mode
of determining the fixed points makes still more. The
best way of marking these points is first to expose the
thermometer for several hours to wet aqueous vapor at
a known pressure, and mark on its stem the height of
the column of mercury. The thermometer is then
brought down to the temperature of melting ice, as
rapidly as possible, and is immersed in pounded ice
which is melting and from which the water is not
allowed to drain off. The mercury being watched
with a magnifying glass is seen to fall, to come to
rest, and to commence to rise, owing to the lagging
contraction of the glass. Its lowest point is marked
on the stem. The interval between the two marks is
then divided into equal degrees. When such a ther-
mometer is used, it is kept at the temperature to be
determined for as long a time as possible, and imme-
diately after is cooled as rapidly as it is safe to cool it,
and its zero is redetermined. Thermometers, so made
and treated, will give very constant indications. But
the thermometers made at the Kew observatory, which
are used for determining the temperature of the yard,
are otherwise constructed. Namely the melting-point
is determined first and the boiling-point afterwards \
and the thermometers are exposed to both tempera-

THE EXTENSION OF THE PRINCIPLES. 283

tures for many hours. The point which upon such a Relative
thermometer will appear as 62° will really be consider- the metre
ably hotter (perhaps a third of a centigrade degree) " ^""^ '
than if its melting-point were marked in the other way.
If this circumstance is hot attended to in making com-
parisons, there is danger of getting the yard too short
by perhaps one two-hundred-thousandth part. General
Comstock finds the metre equal to 39*36985 inches.
Several less trustworthy determinations give nearly the
same value. This makes the inch 2 * 540014 centimetres.

At the time the United States separated from Eng- The Amen-

* can anit of

land, no precise standard of length was legal*; and length,
none has ever been established. We are, therefore,
without any precise legal yard ; but the United States
office of weights and measures, in the absence of any
legal authorisation, refers standards to the British Im-
perial Yard.

The regular British unit of mass is the Pound, de- The British
fined as the mass of a certain platinum weight, called ^m?
the Imperial Pound. This was intended to be so con- _
structed as to be equal to 7000 grains, each the ^6oth
part of a former Imperial Troy pound. This would be
within 3 grains, perhaps closer, of the old avoirdupois
pound. The British pound has been determined by
Miller to be 0-4535926525 kilogramme ; that is the kilo-
gramme is 2-204621249 pounds.

At the time the United States separated from Great
Britain, there were two incommensurable units of
weight, the avoirdupois pound and the Troy pound. Con-
gress has since established a standard Troy pound,
which is kept in the Mint in Philadelphia. It was a
copy of the old Imperial Troy pound which had been
adopted in England after American independence. It

* The ao-called standard of 1758 had not been legalised.

284 THE SCIENCE OF MECHANICS.

TheAmeri- IS a hollow brass Weight of unknown volume : and no

can unit of . ,

mass. accurate comparisons of it with modem standards have
ever been published. Its mass is, therefore, unknown.
The mint ought by law to use this as the standard of
gold and silver. In fact, they use weights furnished
by the office of weights and measures, and no doubt
derived from the British unit ; though the mint officers
profess to compare these with the Troy pound of the
United States, as well as they are able to do. The old
avoirdupois pound, which is legal for most purposes,
differed without much doubt quite appreciably from
the British Imperial pound ; but as the Office of Weights
and Measures has long been, without warrant of law,
standardising pounds according to this latter, the legal
avoirdupois pound has nearly disappeared from use of
late years. The makers of weights could easily detect
the change of practice of the Washington Office.

Measures of capacity are not spoken of here, be-
cause they are not used in mechanics. It may, how-
ever, be well to mention that they are defined by the
weight of water at a given temperature which they
measure.
The unit of The Universal unit of time is the mean solar day or
its one 86400th part, which is called a second. Side-
real time is only employed by astronomers for special
purposes.

Whether the International or the British units are
employed, there are two methods of measurement of
mechanical quantities, the absolute and \}[i^ gravitational.
The absolute is so called because it is not relative to
the acceleration of gravity at any station. This method
was introduced by Gauss.

The special absolute system, widely used by physi-
cists in the United States and Great Britain, is called

THE EXTENSION OF THE PRINCIPLES, 285

the Centimetre-Gramine-Second system. In this sys- The abso-
lute system

tern, writing C for centimetre, G for gramme mass, of the

United

and S for second, states and

Great Brit-

the unit of length is C

the unit of mass is G

the unit of time is S

the unit of velocity is C/S *

the unit of acceleration (which might
be called a "galileo," because Gali-
leo Galilei first measured an accele-
ration) is C/S 2

the unit of density is G/C '

the unit of momentum is G C/S

the unit of force (called a dyne) is ... G C/S ^
the unit of pressure (called one mil-
lionth of an absolute atmosphere) is . . G/C S^ ;
the unit of energy (tns viva, or work,

called an erg) is JGC*/S*;

etc.

The gravitational system of measurement of me- The cravi-
chanical quantities, takes the kilogramme or pound, or system,
rather the attraction of these towards the earth, com-
pounded with the centrifugal force, — which is the ac-
celeration called gravity, and denoted by g, and is dif-
ferent at different places, — as the unit of force, and
the foot-pound or kilogramme-metre, being the amount
of gravitational energy transformed in the descent of a
pound through a foot or of a kilogramme through a
metre, as the unit of energy. Two ways of reconciling
these convenient units with the adherence to the usual
standard of length naturally suggest themselves, namely,
first, to use the pound weight or the kilogramme weight
divided by g as the unit of mass, and, second, to adopt

286 THE SCIENCE OF MECHANICS.

such a unit of time as will make the acceleration of g,
at an initial station, unity. Thus, at Washington, the
acceleration of gravity is 980 • 05 galileos. If, then,
we take the centimetre as the unit of length, and the
0*03 1 943 second as the unit of time, the acceleration
of gravity will be i centimetre for such unit of time
squared. The latter system would be for most pur-
poses the more convenient ; but the former is the more
familiar.
Coropari- In either system, the formula p=zmg is retained:

son of the , J f /- 6 i

absolute, but in the former g retains its absolute value, while in

and gravi- .

tationai the latter it becomes unity for the initial station. In

systems.

Paris, g is 980 '96 galileos ; in Washington it is 980*05
galileos. Adopting the more familiar system, and
taking Paris for the initial station, if the unit of force
is a kilogramme's weight, the unit of length a centi-
metre, and the unit of time a second, then the unit of
mass^will be 1/981-0 kilogramme, and the unit of
energy will be a kilogramme-centimetre, or (1/2)-
(1000 798 1 0)0 0 2/5 a. Then, at Washington the
gravity of a kilogramme will be, not i, as at Paris,
but 980-1/981 •0 = 0-99907 units or Paris kilogramme-
weights. Consequently, to produce a force of one Paris
kilogramme- weight we must allow Washington gravity
to act upon 981 -0/980 •! = i -00092 kilogrammes.]

In mechanics, as in some other branches of physics
closely allied to it, our calculations involve but three
fundamental quantities, quantities of space, quantities
of time, and quantities of mass. This circumstance is
a sourpe of simplification and power in the science
which should not be underestimated.

THE EXTENSION OF THE PRINCIPLES. 287

III.

THE LAWS OF THE CONSERVATION OF MOMENTUM, OF THE

CONSERVATION OF THE CENTRE OF GRAVITY, AND

OF THE CONSERVATION OF AREAS.

1 . Although Newton's principles are fully adequate Spedaiisa-
to deal with any mechanical problem that may arise, mechanical
it is yet convenient to contrive for cases more frequently
occurring, particular rules, which will enable us to treat
problems of this kind by routine forms and to dis-
pense with the minute discussion of them. Newton

and his successors developed several such principles.
Our first subject will be Newton's doctrines concern-
ing freely movable material systems.

2. If two free masses tn and m* are subjected in Mutual ac-
Ihe direction of their line of junction to the action of masses.
forces that proceed from other masses, then, in the in-
terval of time /, the velocities v^ v* will be generated,

and the equation (^p -^ f^ t = m v -{- m'v* will subsist.
This follows from the equations// = wz^ 2ind p'f =
m'v'. The sum mv '\- m'v' is called the momentum of
the system, and in its computation oppositely directed
forces and velocities are regarded as having opposite
signs. If, now, the masses m, m* in addition to being
subjected to the action of the external forces /, /' are
also acted upon by internal forces, that is by such as
are mutually exerted by the masses on one another, these
forces will, by Newton's third law, be equal and op-
posite, q, — q. The sum of the impressed impulses

is, then, (/+/' + ^ — ^)^ = (/ +/')^' ^^® same as
before ; and, consequently, also, the total momentum
of the system will be the same. The momentum of a

288 THE SCIENCE OF MECHANICS.

system is thus determined exclusively by ex/erna/ iorceSf
that isy by forces which masses outside of the system
exert on its parts.
Law of the Imagine a number of free masses »i, m\ m*\ . . .

Cons6rvt&-

Uon or Mo- distributed in any manner in space and acted on by

mentum.

external forces /, /', /". . . . whose lines have any di-
rections. These forces produce in the masses in the
interval of time / the velocities v^ v\ v*\ . . . Resolve
all the forces in three directions jc, y^ z at right angles
to each other, and do the same with the velocities.
The sum of the impulses in the a:-direction will be equal
to the momentum generated in the jc-direction ; and
so with the rest. If we imagine additionally in action
between the masses m^ m*, m". . . ., pairs of equal and
opposite internal forces q, — ^> '', — '', s, — j, etc. ,
these forces, resolved, will also give in every direction
pairs of equal and opposite components, and will con-
sequently have on the sum-total of the impulses no in-
fluence. Once more the momentum is exclusively de-
termined by external forces. The law which states
this fact is called the law of the conservation of momen-
tum.
Law of the 3. Another form of the same principle, which New-

Conserva-
tion of the ton likewise discovered, is called the law of the comer-

Centre of •>•»>-

Gravity. vatton of the centre of grav-

' ^ Ji — ^ ^ p ity. Imagine in A and B

(Fig. 149) two masses, 2w
^^^' '*^* and m, in mutual action,

say that of electrical repulsion ; their centre of gra;^ity
is situated at 5, where BS= 2 AS. The accelerations
they impart to each other are oppositely directed and
in the inverse proportion of the masses. If, then, in
consequence of the mutual action, 2 m describes a dis-
tance AD, m will necessarily describe a distance BC =

THE EXTENSION OF THE PRINCIPLES. 289

ilAD, The point S will still remain the position of the
centre of gravity, as CS = 2DS. Therefore, two masses
cannot, by mutual action, displace their common centre
of gravity.

If our considerations involve several masses, dis- This law

applied to

tributed in any way in space, the same result will also systems of

masses.

be found to hold good for this case. For as no two of
the masses can displace their centre of gravity by mu-
tual action, the centre of gravity of the system as a
whole cannot be displaced by the mutual action of its
parts.

Imagine freely placed in space a system of masses
Pij m\ m'\ . . . acted on by external forces of any kind.
We refer the forces to a system of rectangular co6rdi-
nates and call the coordinates respectively x, y^ z, x\
y, 2', and so forth. The coordinates of the centre of
gravity are then

^ 2mx 2 my 2mz ^

2,m ^m ^ m

in which expressions jc, y^ z may change either by uni-
form motion or by uniform acceleration or by any other
law, according as the mass in question is acted on by
no external force, by a constant external forpe, or by a
variable external force. The centre of gravity will have
in all these cases a different motion, and in the first
may even be at rest. If now internal forces, acting be-
tween every two masses, /// and m*\ come into play in
the system, opposite displacements a/', w" will thereby
be produced in the direction of the lines of junction
of the masses, such that, allowing for signs, m*^/ -(-
»i"w" = 0. Also with respect to the components x^
and x^ of these displacements the equation m*x^ +
m**x^ = 0 will hold. The internal forces consequently

ago THE SCIENCE OF MECHANICS,

produce in the expressions ^, r}y ^ only such additions
as mutually destroy each other. Consequently, the
motion of the centre of gravity of a system is determined
by external forces only.
Acceiera- If we wish to know the acceleration of the centre of

lion of the

centre of gravity of the system, the accelerations of the system's

gravity of a i--ii ^ tr § ** j

system. parts must be similarly treated. If ^, 9? , 9? . . . . de-
note the accelerations of m^ m\ m*\ ... in any direc-
tion, and <p the acceleration of the centre of gravity in
the same direction, <p = 2m(p/^m, or putting the
total mass 2m = Jf, q} = 2m (p/M. Accordingly, we
obtain the acceleration of the centre of gravity of a
system in any direction by taking the sum of all the
forces in that direction and dividing the result by the
total mass. The centre of gravity of a system moves
exactly as if all the masses and all the forces of the
system were concentrated at that centre. Just as a
single mass can acquire no acceleration without the
action of some external force, so the centre of gravity
6f a system can acquire no acceleration without the
action of external forces.

4. A few examples may now be given in illustra-
tion of the principle of the conservation of the centre
of gravity.
Movement Imagine an animal free in space. If the animal
maUree*in move in One direction a portion m of its mass, the re-
space, mainder of it J/ will be moved in the opposite direction,
always so that its centre of gravity retains its original
position. If the animal draw back the mass m, the
motion of M also will be reversed. The animal is un-
able, without external supports or forces, to move itself
from the spot which it occupies, or to alter motions im-
pressed upon it from without.
'"A lightly running vehicle A is placed on rails and

THE EXTENSION OF THE PRINCIPLES. 291

loaded with stones. A man stationed in the vehicle of a ve-

, . 1 -I . hicle, from

casts out the stones one after another, m the same di- which

stones are

rection. The vehicle, supposing the friction to be suf- cast,
ficiently slight, will at once be set In motion in the op-
posite direction. The centre of gravity of the system
as a whble (of the vehicle -|- ^'^ stones) will, so far as
its motion is not destroyed by external obstacles, con-
tinue to remain in its original spot. If the same man
were to pick up the stones from without and place
them in the vehicle, the vehicle in this case would also
be set in motion ; but not to the same extent- as before,
as the following example will render evident.

A projectile of mass ni is thrown with a velocity v Motion of a

"^ ' , •' cannon and

from a cannon of mass M, In the reaction, M also re- it? projec-

tile.

ceives a velocity, F, such that, making allowance for
the signs, MV -\' mv = 0. This explains the so-called
recoil. The relation here is V= — (m/M)v; or, for
equal velocities of flight, the recoil is less according as
the mass of the cannon is greater than the mass of the
projectile. If the work done by the powder be expressed
by A, the vt'res viva will be determined by the equation
MV^/2 -{- mv* /2 = A ; and, the sum of the momenta
being by the first-cited equation = 0, we readily obtain
y=\/2Am/M(^M-{- m). Consequently, neglecting
the mass of the exploded powder, the recoil vanishes
when the mass of the projectile vanishes. If the mass
m were not expelled from the cannon but sucked into
it, the recoil would take place in the opposite direc-
tion. But it would have no time to make itself visible
since before any perceptible distance had been trav-
ersed, m would have reached the bottom of the bore.
As soon, however, as Af and m are in rigid connection
with each other, as soon, that is, a^ they are relatively
at rest to each other, they must be absolutely at rest.

292 THE SCIENCE OF MECHANICS.

for the centre of gravity of the system as a whole has
no motion. For the same reason no considerable mo-
tion can take place when the stones in the preceding
example are taken into the vehicle, because on the
establishment of rigid connections between the vehicle
and the stones the opposite momenta generated are
destroyed. A cannon sucking in a projectile would
experience a perceptible recoil only if the sucked in
projectile could fly through it.
osciiia- Imagine a locomotive freely suspended in the air,

tionsof the i .n i i • i

body of a or, what Will subserve the same purpose, at rest with

locomotive. . ^^. r-- i •» -r^ii r \

insufficient friction on the rails. By the law of the
conservation of the centre of gravity, as soon as the
heavy masses of iron in connection with the piston-
rods begin to oscillate, the body of the locomotive will
be set in oscillation in a contrary direction — a motion
which may greatly disturb its uniform progress. To
eliminate this oscillation, the motion of the masses of
iron worked by the piston-rods must be so compensated
for by the contrary motion of other masses that the
centre of gravity of the system as a whole will remain
in one position. In this way no motion of the body of
the locomotive will take place. This is done by affix-
ing masses of iron to the driving-wheels.
Illustration The facts of this case may be very prettily shown
case. by Page's electromotor (Fig. 150). When the iron

core in the bobbin -^^ is projected by the internal forces
acting between bobbin and core to the right, the body
of the motor, supposing it to rest on lightly movable
wheels rr^ will move to the left. But if to a spoke of
the fly-wheel R we affix an appropriate balance-weight
a, which always moves in the contrary direction to the
iron core, the sideward movement of the body of the
motor may be made totally to vanish.

THE EXTENSION OF THE PRINCIPLES. 293

Of the motion of the fragments of a bursting bomb a bm
we know nothing. But it is plain, by the law of the
conservation of the centre of gravity, that, making al-
lowance for the resistance of the air and the obstacles
the individual parts may meet, the centre of gravity of
the system will continue after the bursting to describe
the parabolic path of its original projection.

5. A law closely allied to the law of the centre of l«wi
gravity, and similarly applicable io/ree systems, is theHont
principle of the amservation of areas. Although Newton

had, so to say, this principle within his very grasp, it
was nevertheless not enunciated until a long time after-
wards by EuLER, D'Arcy, and Daniel Bernoulli.
Euler and Daniel Bernoulli discovered the law almost
simultaneously (1746), on the occasion of treating a
problem proposed by Euler concerning the motion of
balls in rotatable tubes, being led to it by the consider-
ation of the action and reaction of the balls and the
tubes. D'Arcy (1747) started from Newton's investiga-
tions, and generalised the law of sectors which the
latter had employed to explain Kepler's laws.

294

THE SCIENCE OF MECHANICS.

Deduction
of the law.

Two masses m, ni (Fig. 151) are in mutual action.
By virtue of this action the masses describe the dis-
tances ABy CD in the direction of their line of junction.
Allowing for the signs, then, m . AB + m*, CD = 0.
Drawing radii vectores to the moving masses from any

point Oy and regarding
the areas described in
opposite senses by the
T radii as having opposite

signs, we further obtain
^^ m.OAB + m\OCD = Q.
Which is to say, if two
masses mutually act on
each other, and radii vec-
tores be drawn to these
masses from any point,
the sum of the areas
described by the radii
multiplied by the respec-
If the masses are also acted on
by external forces and as the effect of these the areas
OAE and OCF are described, the joint action of the
internal and external forces, during any very small
period of time, will produce the areas OAG^xnA OCH,
But it follows from Varignon's theorem that

mOAG + m'OCH= m OAE + m OCF +
mOAB + niOCD = mOAE + m'OCF;

in other words, the sum of the products of the areas so de-
scribed into the respective masses which compose a system
is unaltered by the action of internal forces.

If we have several masses, the same thing may be
asserted, for every two masses, of the projection on any
given plane of the motion. If we draw radii from

tive masses is = 0.

THE EXTENSION OF THE PRINCIPLES. 295

any point to the several masses, and project on any
plane the areas the radii describe, the sum of the
products of these areas into the respective masses will
be independent of the action of internal forces. This
is the law of the conservation of areas.

If a single mass not acted on by forces is moving interpretn-

, , tion of the

uniformly forward in a straight line and we draw a law.
radius vector to the mass from any point (9, the area
described by the radius increases proportionally to the
time. The same law holds for ^mf in cases in which
several masses not acted on by forces are moving,
where we signify by the summation the algebraic sum
of all the products of the areas (/) into the moving
masses — a sum which we shall hereafter briefly refer
to as the sum of the mass-areas. If internal forces
come into play between the masses of the system, this
relation will remain unaltered. It will still subsist,
also, if external forces be applied whose lines of action
pass through ih& fixed point Oy as we know from the
researches of Newton.

If the mass be acted on by an external force, the
area / described by its radius vector will increase in
time by the law/= at^ ji -{- bt -\- c^ where a depends
on the accelerative force, b on the initial velocity, and
c on the initial position. The sum ^mf increases by
the same law, where several masses are acted upon by
external accelerative forces, provided these may be Re-
garded as constant, which for sufficiently small inter-
vals of time is always the case. The law of areas in
this case states that the internal forces of the system
have no influence on the increase of the sum of the mass-
areas.

A free rigid body may be regarded as a system
whose parts are maintained in their relative positions

296 THE SCIENCE OF MECHANICS.

Uniform ro- bv internal forces. The law of areas is applicable there -

tation of a ,

free rigid fore to this case also. A simple instance is afforded

body. . ^

by the uniform rotation of a rigid body about an axis
passing through its centre of gravity. If we call m a
portion of its mass, r the distance of the portion from
the axis, and a its angular velocity, the sum of the
mass-areas produced in unit of time will be ^m
{r/2)ra = (^a/2)2mr^, or, the product of the moment
of inertia of the system into half its angular velocity.
This product can be altered only by external forces.
Illustrative 6. A few examples may now be cited in illustration

examples, ^f ^^e kw.

If two rigid bodies ^ and K' are connected, and IC
is brought by the action of internal forces into rotation
relatively to IC\ immediately A" also will be set in ro-
tation, in the opposite direction. The rotation of AT
generates a sum of mass-areas which, by the law, must
be compensated for by the production of an equals but
opposite, sum by K\
Opposite This is very prettily exhibited by the electromotor

the wheel of Fig. 1 5 2. The fly-wheel of the motor is placed in
a^free^eiec'- a horizontal plane, and the motor thus attached to a
vertical axis, on which it can freely turn. The wires
conducting the current dip, in order to prevent their
interference with the rotation, into two conaxial gutters
of mercury fixed on the axis. The body of the motor
{X') is tied by a thread to the stand supporting the
axis and the current is turned on. As soon as the fly-
wheel (-^), viewed from above, begins to rotate in the
direction of the hands of a watch, the string is drawn
taut and the body of the motor exhibits the tendency
to rotate in the opposite direction — a rotation which im-
mediately takes place when the thread is burnt away.
The motor is, with respect to rotation about its

THE EXTENSION OF THE PRINCIPLES. ^97

axis, a free system. The sum of the mass-areas gen- i.i>e
erated, for the case of rest, is ^ 0. But the -wheel of law.
the motor being set in rotation by the action of the in-
ternal electro -magnetic forces, a sum of mass-areas is

produced which, as the total sum must remain ^ 0, is
compensated for by the rotation in the opposite direc-
tion of the body of the motor, ff an index be attached
to the body of the motor and kept in a fixed position

non.

2^ THE SCIENCE OF MECHANICS,

by an elastic spring, the rotation of the body of the
motor cannot take place. Yet every acceleration of
the wheel in the direction of the hands of a watch (pro-
duced by a deeper immersion of the battery) causes
the index to swerve in the opposite direction, and every
retardation produces the contrary effect,
A variation A beautiful but curious phenomenon presents itself
pheno^!** when the current to the motor is interrupted. Wheel
and motor continue at first their movements in oppo-
site directions. But the effect of the friction of the
axes soon becomes apparent and the parts gradually
assume with respect to each other relative rest. The
motion of the body of the motor is seen to diminish ;
for a moment it ceases ; and, finally, when the state of
relative rest is reached, it is reversed and assumes the
direction of the original motion of the wheel. The
whole motor now rotates in the direction the wheel did
at the start. The explanation of the phenomenon is
obvious. The motor is not a perfectly free system. It
is impeded by the friction of the axes. In a perfectly
. free system the sum of the mass-areas, the moment
the parts re-entered the state of relative rest, would
again necessarily be = 0. But in the present instance,
an external force is introduced — the friction of the
axes. The friction on the axis of the wheel diminishes
the mass-areas generated by the wheel and body of
the motor alike. But the friction on the axis of the
body of the motor only diminishes the sum of the mass-
areas generated by the body. The wheel retains, thus,
an excess of mass- area, which when the parts are rela-
tively at rest is rendered apparent in the motion of the
entire motor. The phenomenon subsequent to the in-
terruption of the current supplies us with a model of
what according to the hypothesis of astronomers has

THE EXTENSION OF THE PRINCIPLES. 299

taken place on the moon. The tidal wave created by its iiiostra-
the earth has reduced to such an extent by friction the case of the
velocity of rotation of the moon that the lunar day has
grown to a month. The fly-wheel represents the fluid
mass moved by the tide.

Another example of this law is furnished by reac- Reaction-

wheels.

tion-wheeis. If air or gas be emitted from the wheel
(Fig. 153^) in the direction of the short arrows, the
whole wheel will be set in rotation in the direction of
the large arrow. In Fig. 153^, another simple reac-
tion-wheel is represented. A brass tube rr plugged at
both ends and appropriately perforated, is placed on a
second brass tube R, supplied with a thin steel pivot
through which air can be blown ; the air escapes at
the apertures (9, ff.

It might be supposed that sucking on the reaction- variation

i_ 1 ij J t . . t ofthephe-

wheels would produce the opposite motion to that re- nomcna of
suiting from blowing. Yet this does not usually take wheels,
place, and the reason is obvious. The air that is
sucked into the spokes of the wheel must take part
immediately in the motion of the wheel, must enter
the condition of relative rest with respect to the wheel ;
and when the system is completely at rest, the sum of
its mass-areas must be =: 0. Generally, no perceptible
rotation takes place on the sucking in of the air. The
circumstances are similar to those of the recoil of a
cannon which sucks in a projectile. If, therefore, an
elastic ball, which has but one escape- tube, be attached
to the reaction-wheel, in the manner represented in
Fig. 153 a, and be alternately squeezed so that the
same quantity of air is by turns blown out and sucked
in, the wheel will continue rapidly to revolve in the
same direction as it did in the case in which we blew
into it. This is partly due to the fact that the air

300 THE SCIENCE OF MECHANICS.

THE EXTENSION OF THE PRINCIPLES. 301

sucked into the spokes must participate in the motion Ezpiana-
of the latter and therefore can produce no reactional variations,
rotation, but it also partly results from the difference
of the motion which the air outside the tube assumes
in the two cases. In blowing, the air flows out in jets,
and performs rotations. In sucking, the air comes in
from all sides, and has no distinct rotation.

The correctness of this view is easily demonstrated.
If we perforate the bottom of a hollow cylinder, a closed
band-box for instance, and
place the cylinder on the steel
pivot of the tube R^ after the
side has been slit and bent in
the manner indicated in Fig.
154, the box will turn in the
direction of the long arrow
when blown into and in the fir. 154-

direction of the short arrow when sucked on. The air,
here, on entering the cylinder, can continue its rotation
unimpeded^ and this motion is accordingly compensated
for by a rotation in the opposite direction.

7. The following case also exhibits similar condi- ReacUon-
tions. Imagine a tube (Fig. 155a) which, running
straight from a to ^, turns at right angles
to itself at the latter point, passes to r,
describes the circle cdef^ whose plane
is at right angles to ab^ and whose cen-
tre is at by then proceeds from / to g^
and, finally, continuing the straight line
aby runs from g to h. The entire tube
is free to turn on an axis ah. If we
pour into this tube, in the manner in-
dicated in Fig. 155^, a liquid, which flows in the di-
rection cdefy the tube will immediately begin to turn

3oa THE SCIENCE OF MECHANICS.

in the direction /ir^i-. This impulse, however, ceases,
the moment the liquid reaches the point _;^ and Bowing
out into the radius/g- is obliged to join in the motion
of the latter. By the use of a constant stream of liquid,
therefore, the rotation
of the tube may soon
be stopped. But if the
stream be interrupted,
the fluid, in flowing off
through the radius fg,
will impart to the tube
a motional impulse in
the direction of its own
motion, cdef, and the
tube will turn in this di-
rection. All these phe-
nomena are easily ex-
plained by the law of

The trade- winds, the
deviation of the oceanic
currents and of rivers,
Foucault's pendulum
experiment, and the
like,*mayalso be treated
'''*■ '"''■ as examples of the law

1 of areas. Another pretty illustration is afforded by
bodies with variable moments of inertia. Let a body
with the moment of inertia & rotate with the angular
velocity a and, during the motion, let its moment
of inertia be transformed by internal forces, say by
springs, into &, a will then pass into a", where a© =
a'©', that is a' = a{&j&'). On any considerable dimi-
nution of the moment of inertia, a great increase of

THE EXTENSION OF THE PRINCIPLES. 303

angular velocity ensues. The principle might con-
ceivably be employed, instead of Foucault's method,
to demonstrate the rotation of the earth, [in fact, some
attempts at this have been made, with no very marked
success] .

A phenomenon which substantially embodies the RotatinR
conditions last suggested is the following. A glass funnel,
funnel, with its axis placed in a vertical position, is
rapidly filled with a liquid in such a manner that the
stream does not enter in the direction of the axis but
strikes the sides. A slow rotatory motion is thereby
set up in the liquid which as long as the funnel is full, is
not noticed. But when the fluid retreats into the neck
of the funnel, its moment of inertia is so diminished
and its angular velocity so increased that a violent
eddy with considerable axial depression is created.
Frequently the entire effluent jet is penetrated by an
axial thread of air.

8. If we carefully examine the principles of the Both pnn-

, . ciples are

centre of gravity and of the areas, we shall discover in simply spe-

, , cialcasesof

both simply convenient ^ -,^ the law of

^ ^ . SQL 2cg^ action and

modes of expression, for Jt. V-i2w n^ [ ?2^ reaction,
practical purposes, of
a well-known property
of mechanical phenom-
ena. To the accelera-
tion q} of one mass m ^**- ^^
there always corresponds a contrary acceleration <ff of
a second mass tn\ where allowing for the signs tnq}'\-
m' <p' = 0. To the force m (p corresponds the equal
and opposite force m'<p\ When any masses m and
2 m describe with the contrary accelerations 2 (p and (p
the distances iw and w (Fig. 156), the position of
their centre of gravity S remains unchanged, and the

304 THE SCIENCE OF MECHANICS.

sum of their mass-areas with respect to any point O
is, allowing for the signs, 2w./+w.2/=0. This
simple exposition shows us, that the principle of the
centre of gravity expresses the same thing with respect
to parallel coordinates that the principle of areas ex-
presses with respect to polar coordinates. Both contain
simply the fact of reaction.
But they The principles in question admit of still another

construed Simple coustructiou. Just as a single body cannot,
satfons%'f ' without the influence of external forces, that is, without
inertia. the aid of a second body, alter its uniform motion of
progression or rotation, so also a system of bodies can-
not, without the aid of a second system, on which it
can, so to speak, brace and support itself, alter what
may properly and briefly be called its mean velocity of
progression or rotation. Both principles contain, thus,
a generalised statement of the law of inertia, the correct-
ness of which in the present form we not only see but
feel.
Importance This feeling is not unscientific; much less is it
stinctive detrimental. Where it does not replace conceptual in-
mechanicai sight but exists by the side of it, it is really the funda-

facts*

mental requisite and sole evidence of a complete mastery
of mechanical facts. We are ourselves a fragment of
mechanics, and this fact profoundly modifies our mental
life. * No one will convince us that the consideration
of mechanico-physiological processes, and of the feel-
ings and instincts here involved, must be excluded from
scientific mechanics. If we know principles like those
of the centre of gravity and of areas only in their ab-
stract mathematical form, without having dealt with the
palpable simple facts, which are at once their applica-

* For the development of this view, see E. Mach, Grundlinien d*r Lekrt.
von den Bewegungsfm/ifindunfen, (Leipsic : Engelmann, 1875.)

THE EXTENSION OF THE PRINCIPLES. 305

tion and their source, we only half comprehend them,
and shall scarcely recognise actual phenomena as ex-
amples of the theory. We are in a position like that
of a person who is suddenly placed on a high tower
but has not previously travelled in the district round
about, and who therefore does not know how to inter-
pret the objects he sees.

IV.
THE LAWS OF IMPACT.

I. The laws of impact were the occasion of the Historical

, position of

enunciation of the most important principles of me- the Laws of
chanics, and furnished also the first examples of the
application of such principles. As early as 1639, a
contemporary of Galileo, the Prague professor, Marcus
Marci (bom in 1595), published in his treatise De Pro-
portione Motus (Prague) a few results of his investiga-
tions on impact. He knew that a body striking in
elastic percussion another of the same size at rest, loses
its own motion and communicates an equal quantity
to the other. He also enunciates, though not always
with the requisite precision, and frequently mingled
with what is false, other propositions which still hold
good. Marcus Marci was a remarkable man. He pos-
sessed for his time very creditable conceptions regard-
ing the composition of motions and "impulses." In
the formation of these ideas he pursued a method sim-
ilar to that which Roberval later employed. He speaks
of partially equal and opposite motions, and of wholly
opposite motions, gives parallelogram constructions,
and the like, but is unable, although he speaks of an
accelerated motion of descent, to reach perfect clear-
ness with regard to the idea of force and consequently
also with regard to the composition of forces. In spite

3o6 rim SCIENCE OF MECHANICS.

ere. of this, howBver, he discovers Galileo's theorem re-
rciii garding the descent of bodies in the chords of circles,

also a few propositions relating to the motion of the
pendulum, and has knowledge of centrifugal force and
so on. Although Galileo's Discourses had app>eared a

THE EXTENSION OF THE PRINCIPLES.

year previously, we cannot, in view of the condition of
things produced in Central Eutope by the Thirty Years'

War, assume that Marci was acquainted with them.
Not only would the many errors in Marci's book thus
be rendered unintelligible, but it would also have to

3o8 THE SCIENCE OF MECHANICS,

The sources be explained how Marci, as late as 1648, in a continu-

of Marci's , » • . . « « < , * .

knowledge, ation of his treatise, could have found it necessary to
defend the theorem of the chords of circles against the
Jesuit Balthasar Conradus. An imperfect oral com-
munication of Galileo's researches is the more reason-
able conjecture.* When we add to all this that Marci
was on the very verge of anticipating Newton in the
discovery of the composition of light, we shall recog-
nise in him a man of very considerable parts. His
writings are a worthy and as yet but slightly noticed
object of research for the historian of physics. Though
Galileo, as the clearest-minded and most able of his
contemporaries, bore away in this province the palm,
we nevertheless see from writings of this class that he
was not by any means alone in his thought and ways
of thinking.
There- 2. Galileo himsclf made several experimental at-

Galileo, tempts to ascertain the laws of impact ; but he was not
in these endeavors wholly successful. He principally
busied himself with the force of a body in motion, or
with the "force of percussion," as he expressed it,
and endeavored to compare this force with the pressure
of a weight at rest, hoping thus to measure it. To this
end he instituted an extremely ingenious experiment,
which we shall now describe.

A vessel I (Fig. 157) in whose base is a plugged
orifice, is filled with water, and a second vessel H is
hung beneath it by strings ; the whole is fastened to
the beam of an equilibrated balance. If the plug is
removed from the orifice of vessel I, the fluid will fall

* I have been convinced, since the publication of the first edition of this
work, (see E. Wohlwill's researches, Die Entdeckung de» Beharmngtgtutzes,
in the Zeitsckri/ffUr VdVterptyckoUgie^ 1884, XV, page 387,) that Marcus Marci
derived his information concerning the motion of falling bodies, from Galileo's
tarlitr Dialogues. — AtUhar*s Appendix te Second Edition.

THE EXTENSION OF THE PRINCIPLES. 309

in a jet into vessel II. A portion of the pressure due Caiiiao'i
to the resting weight of the water in I is lost and re- man.
placed by an action of impact on vessel II. Galileo
expected a depression of the whole scale, by which he
hoped with the assistance of a counter- weight to de-
termine the effect of the impact. He was to some ex-
tent surprised to obtain no depression, and he was un-
able, it appears, perfectly to clear up the matter in his
mind.

3. To-day, of course, the explanation is not diffi-
cult. By the removal of the plug there is produced,

first, a diminution of the pressure. This consists of EipUna-
two factors: (1) The weight of the jet suspended ineiperi-
the air is lost ; and (2) A reaction-pressure upwards is
exerted by the efHueut jet on vessel I (which acts like
a Segner's wheel). Then there is an increase of pres-
sure (Factor 3) produced by the action of the jet on the
bottom of vessel II. Before the first drop has reached
the bottom of II, we have only to deal with a diminu-
tion of pressure, which, when the apparatus is in full
operatjion, is immediately compensated for. 'Y\\\% initial

3IO THE SCIENCE OF MECHANICS,

Deterraina- depression was, in fact, all that Galileo could observe.

tion of the ^ • • t • • 11

mechanical LrCt US imagine the apparatus m operation, and denote
voived. the height the fluid reaches in vessel I by //, the corre-
sponding velocity of efflux by r, the distance of the
bottom of I from the surface of the fluid in II by >t, the
velocity of the jet at this surface by w, the area of the
basal orifice by a, the acceleration of gravity by gy and
the specific gravity of the fluid by s. To determine
Factor (i) we may observe that v is the velocity ac-
quired in descent through the distance h. We have,
then, simply to picture to ourselves this motion of de-
scent continued through k. The time of descent of
the jet from I to II is therefore the time of descent
through h '\' k less the time of descent through /i.
During this time a cylinder of base a is discharged
with the velocity v. Factor (i), or the weight of the
jet suspended in the air, accordingly amounts to

^^{m'^-m

as.

To determine Factor (2) we employ the familiar
equation piv =^/. If we put / = i, then pt7f =p, that
is the pressure of reaction upwards on I is equal to the
momentum imparted to the fluid jet in unit of time.
We will select here the unit of weight as our unit of
force, that is, use gravitation measure. We obtain for
Factor (2) the expression [^ v {s/g)'] v = p, (where the
expression in brackets denotes the mass which flows
out in unit of time,) or

av^2g/i . - . \/2g/i =2ahs.
i

Similarly we find the pressure on II to be

(

av . ^^\w = q, ox factor 3 :
g

THE EXTENSION OF THE PRINCIPLES. 311

J Mathemat'

a VtghVtg {li + /')• '^Zt

^ toe result.

The total variation of the pressure is accordingly

-M^^P-4vh

2a/ts

or, abridged,

— 2as[V^/i(^/i + J^) — //] — 2aAs

+ 2asl/7i{/i + J^,

— which three factors completely destroy each other. In
the very necessity of the case, therefore, Galileo could
only have obtained a negative result.

We must supply a brief comment respecting Fac- a comment

sufjgested

tor (2). It might be supposed that the pressure on the by the ex-

penmeiita

basal orifice which is lost, \^ ahs and not 2ahs, But
this statical conception would be totally inadmissible
in the present, dynamical case. The velocity v is not
generated by gravity instantaneously in the effluent
particles, but is the outcome of the mutual pressure
between the particles flowing out and the particles left
behind ; and pressure can only be determined by the
momentum generated. The erroneous introduction of
the value ahs would at once betray itself by self-con-
tradictions.

If Galileo's mode of experimentation had been less
elegant, he would have determined without much diffi-
culty the pressure which a continuous fluid jet exerts.
But he could never, as he soon became convinced,
have counteracted by a pressure the effect of an instan-
taneous impact. Take — and this is the supposition of

312 THE SCIENCE OF MECHANICS.

Galileo's Galileo — a freely falling, heavy body. Its final veloc-

reasoning. ... • i , .

ity, we know, increases proportionately to the time.
The very smallest velocity requires a definite portion
of time to be produced in (a principle which even Mari-
otte contested). If we picture to ourselves a body
moving vertically upwards with a definite velocity, the
body will, according to the amount of this velocity,
ascend a definite time, and consequently also a definite
distance. The heaviest imaginable body impressed
in the vertical upward direction with the smallest im-
aginable velocity will ascend, be it only a little, in
opposition to the force of gravity. If, therefore, a
heavy body, be it ever so heavy, receive an instan-
taneous upward impact from a body in motion, be the
mass and velocity of that body ever so small, and such
impact impart to the heavier body the smallest imagin-
able velocity, that body will, nevertheless, yield and
Compari- move somewhat in the upward direction. The sHMest

son of the . .

ideas im- impact, therefore, is able to overcome the greatest pres-

pact and . .

pressure, sure ; or, as Galileo says, the force of percussion com-
pared with the force of pressure is infinitely great. This
result, which is sometimes attributed to intellectual ob-
scurity on Galileo's part, is, on the contrary, a bril-
liant proof of his intellectual acumen. We should say
to-day, that the force of percussion, the momentum,
the impulse, the quantity of motion m ?;, is a quantity
of different dimensions from the pressure /. The dimen-
sions of the former are mlt'^^, those of the latter mlt'~''^.
In reality, therefore, pressure is related to momentum
of impact as a line is to a surface. Pressure is /, the
momentum of impact is / /. Without employing mathe-
matical terminology it is hardly possible to express the
fact better than Galileo did. We now also see why it
is possible to measure the impact of a continuous fluid

THE EXTENSION OF THE PRINCIPLES. 313

jet by a pressure. We compare the momentum de-
stroyed per second of time with the pressure acting
per second of time, that is, homogeneous quantities of
the form//.

4. The first systematic treatment of the laws of The syste-
matic treat-
impact was evoked in the year 1668 by a request of the ment of the

laws of iin-

Royal Society of London. Three eminent physicists pact.
Wallis (Nov. 26, 1668), Wren (Dec. 17, 1668), and
HuYGENS (Jan. 4, 1669) complied with the invitation of
the society, and communicated to it papers in which,
independently of each other, they stated, without de-
ductions, the laws of impact. Wallis treated only of
the impact of inelastic bodies. Wren and Huygens only
of the impact of elastic bodies. Wren, previously to
publication, had tested by experiments his theorems,
which, in the main, agreed with those of Huygens.
These are the experiments to which Newton refers in
the Principia, Thfe same experiments were, soon after
this, also described, in a more developed form, by Ma-
riotte, in a special treatise, Sur le Choc des Corps, Ma-
riotte also gave the apparatus now known in physical
collections as the percussion-machine.

According to Wallis, the decisive factor in impact waiiis's re-
is momentum^ or the product of the mass {pondus) into
the velocity {celeritas). By this momentum the force
of percussion is determined. If two inelastic bodies
which have equal momenta strike each other, rest will
ensue after impact. If their momenta are unequal,
the difference of the momenta will be the momentum
after impact. If we divide this momentum by the sum
of the masses, we shall obtain the velocity of the mo-
tion after the impact. Wallis subsequently presented
his theory of impact in another treatise, Mechanica swe
de MotUy London, 1671. All his theorems may be

314 THE SCIENCE OF MECHANICS.

brought together in the formula now in common use,
u =: {mv -\- m'v')/{m + m"), in which m, m' denote the
masses, i', v' the velocities before impact, and » the
velocity after impact.
■n>'i 5, Tbe ideas which led Huygens to his results, are

Bsiii. to be found in a posthumous treatise of his, De Motu
Corporum ex Percusswne, 1703. We shall examine these
in some detail. The assumptions from which Huygens

0.4 ^^^

proceeds are: (1) the law of inertia; (3) that elastic
bodies of equal mass, colliding with equal and oppo-
site velocities, separate after impact with the same ve-
locities ; (3) that all velocities are relatively estimated ;
(4) that a larger body striking a smaller one at rest
imparts to the latter velocity, and loses a part of its
own ; and finally (5) that when one of the colliding
bodies preserves its velocity, this also is the case with
the other.

THE EXTENSION OF THE PRINCIPLES, 315

Huygens, now, imagines two equal elastic masses, First, equal

clastic

which meet with equal and opposite velocities v. After masses ex-
the impact they rebound from each other with exactly locUies.
the same velocities. Huygens is right in assuming and
not deducing this. That elastic bodies exist which re-
cover their forni after impact, that in such a transac-
tion no perceptible vis viva is lost, are facts which ex-
perience alone can teach us. Huygens, now, conceives
the occurrence just described, to take place on a boat
which is moving with the velocity v. For the specta-
tor in the boat the previous case still subsists ; but for
the spectator on the shore the velocities of the spheres
before impact are respectively 2 v and 0, and after im-
pact 0 and 2 V, An elastic body, therefore, impinging
on another of equal mass at rest, communicates to the
latter its entire velocity and remains after the impact
itself at rest. If we suppose the boat affected with any
imaginable velocity, u^ then for the spectator on the
shore the velocities before impact will be respectively
u -\- V and u — v, and after impact u — v and u -(- v.
But since u -{- v and u — v may have any values what-
soever, it may be asserted as a principle that equal
elastic masses exchange in impact their velocities.

A body at rest, however great, is set in motion Second, the

, relative ve-

by a body which strikes it, however small; as Ga-»ocityofap-

, , proach and

lileo pointed out. Huygens, now, recession is

j^ the same.

shows, that the approach of the «
bodies before impact and their ^

recession after impact take place *

with the same relative velocity. A ^^^' ^^

body m impinges on a body of mass M at rest, to which
it imparts in impact the velocity, as yet undetermined,
w. Huygens, in the demonstration of this proposition,
supposes that the event takes place on a boat moving

3i6 THE SCIENCE OF MECHANICS.

from M towards m with the velocity w/2. The initial
velocities are, then, v — wJT, and — wji, ; and the final
velocities, x and -f~ ^Z^- ^ut as M has not altered
the value, but only the sign, of its velocity, so »», if a
loss of vis viva is not to be sustained in elastic impact,
can only alter the sign of its velocity. Hence, the final
velocities are — {v — w/2) and + wji. As a fact,
then, the relative velocity of approach before impact
is equal to the relative velocity of separation after im-
pact. Whatever change of velocity a body may suffer,
in every case, we can, by the fiction of a boat in mo-
tion, and apart from the algebraical signs, keep the
value of the velocity the same before and after impact.
The proposition holds, therefore, generally.
Third.if the If two masscs M and m collide, with velocities V

VfiloCltlfiS

of approach and v inversely proportional \.o the masses, J/ after im-
ly propor-^ pact will rebouud with the velocity V and m with the
massesrso Velocity V. Let us suppose that the Velocities after
focitie^s oT impact are F^ and Vy^ ; then by the preceding proposi-
tion we must have F'+ v =^V^ + z^j, and by the prin-
ciple of vis viva

MV^ mv^ _MV^^ mv^^
~"2 ^ T"~~ "^"~+ "2"*

Let us assume, now, that v^ =^v -\- w; then, neces-
sarily, V^ = F — w; but on this supposition

J/r2 , mv,^ MV^ . mv^ . ^ ., . . w*
-^-+-2 ="^-+'-2~+<^^+^)"2-

And this equality can, in the conditions of the case,
only subsist if «/ = 0 ; wherewith the proposition above
stated is established.

Huygens demonstrates this by a comparison, con-
structively reached, of the possible heights of ascent
of the bodies prior and subsequently to impact. If

THE EXTENSION OF THE PRINCIPLES, 317

the velocities of the impinging bodies are not inversely This propo-
proportional to the masses, they may be made such by the fiction
the fiction of a boat in motion. The proposition thus boat, made
includes all imaginable cases. aVcaUs/

The conservation of vis viva in impact is asserted
by Huygens in one of his last theorems (ii), which he
subsequently also handed in to the London Society.
But the principle is unmistakably at the foundation of
the previous theorems.

6. In taking up the study of any event or phenom- Typical

modes of

enon Ay we may acquire a knowledge of its component natural in-
elements by approaching it from the point of view of a
different phenomenon B^ which we already know ; in
which case our investigation of A will appear as the
application of principles before familiar to us. Or, we
may begin our investigation with A itself, and, as na-
ture is throughout uniform, reach the same principles
originally in the co^Uemplation of A, The investiga-
tion of the phenomena of impact was pursued simul-
taneously with that of various other mechanical pro-
cesses, and both modes of analysis were really pre-
sented to the inquirer.

To begin with, we may convince ourselves that the impact in

. the New-

problems of impact can be disposed of by the New- tonian

tonian principles, with the help of only a minimum of view.
new experiences. The investigation of the laws of im-
pact contributed, it is true, to the discovery of New-
ton's laws, but the latter do not rest solely on this foun-
dation. The requisite new experiences, not contained
in the Newtonian principles, are simply the informa-
tion that there are elastic and inelastic bodies. Inelastic
bodies subjected to pressure alter their form without
recovering it ; elastic bodies possess for all their /^rwj
definite systems of pressures, so that every alteration

3i8

THE SCIENCE OF MECHANICS.

First, in-
elastic
masses.

of form is associated with an alteration of pressure, and
vice versa. Elastic bodies recover their form ; and the
forces that induce the form -alterations of bodies do not
come into play until the bodies are in contact.

Let us consider two inelastic masses M and m mov-
ing respectively with the velocities V and v* If these
masses come in contact while possessed of these un-
equal velocities, internal form-altering forces will be
set up in the system M, tn. These forces do not alter
the quantity of motion of the system, neither do they
displace its centre of gravity. With the restitution of
equal velocities, the form -alterations cease and in in-
elastic bodies the forces which produce the alterations
vanish. Calling the common velocity of motion after
impact «, it follows that Mu -\- mu = MV-\- Mv, or
u = (MF+ mv)/(Af+ m), the rule of Wallis.
Impact in Now let US assume that we are investigating the

lent point phenomena of impact without a previous knowledge of
Newton's principles. We very soon discover, when
we so proceed, that velocity is not the so/e determina-
tive factor of impact ; still another physical quality is
decisive — weight, load, msiss, pon/fus, moles y massa. The
moment we have noted this fact, the simplest case is
easily dealt with. If two bodies of equal weight or

equal mass collide with equal and
opposite velocities ; if, further, the
bodies do not separate after impact
but retain some common velocity,
plainly the sole uniquely deter-
mined velocity after the collision is the velocity 0. If,
further, we make the observation that only the dif-
ference of the velocities, that is only relative velocity,
determines the phenomenon of impact, we shall, by
imagining the environment to move, (which experience

O O

Fig. x6x.

THE EXTENSION OF THE PRINCIPLES,

319

tells US has no influence on the occurrence,) also readily
perceive additional cases. For equal inelastic masses
with velocities v and 0 or «; and v* the velocity after
impact is vji or {v + v*)!'!. It stands to reason that
we can pursue such a line of reflection only after ex-
perience has informed us what the essential and de-
cisive features of the phenomena are.

If we pass to unequal masses, we must not only The expe-

1 t .1 „ • r riendaf

know from experience that mass generally is of conse- conditions

of this

quence, but also in what manner its influence is eflec- method,
tive. If, for example, two bodies of masses i and 3
with the velocities v and F collide, we might reason

Fig. 163.

thus. We cut out of the mass 3 the mass i (Fig. 162),
and first make the masses i and i collide : the result-
ant velocity is {v + O/^- There are now left, to
equalise the velocities (z^ + F)/2 and V^ the masses
I -f I = 2 and 2, which applying the same principle
gives

-+ V

2 ■" 4 "^ 1 + 3 •

Let us now consider, more generally, the masses
m and fn\ which we represent in Fig. 163 as suitably
proportioned horizontal lines. These masses are af-
fected with the velocities v and v\ which we represent
by ordinates erected on the mass-lines. Assuming that

320

THE SCIENCE OF MECHANICS,

Its points of /^ < m*y we cut off from m' a portion tn. The offsetting
with the of /// and m gives the mass 7.m with the velocity {v +
' e'')/2. The dotted line indicates this relation. We
proceed similarly with the remainder m* — m. We cut
off from 7.m 2l portion m' — w, and obtain the mass
7,m — {m' — ni) with the velocity (7^ + t'')/2 and the
mass 2(/;/' — ni) with the velocity \{v + Z'')/2 + z/']/2.
In this manner we may proceed till we have obtained
for the whole mass m -\' tn* the same velocity u. The
constructive method indicated in the figure shows very
plainly that here the surface equation (m + «r') u =
mv -\' m'v' subsists. We readily perceive, however,
that we cannot pursue this line of reasoning except the
sum mv -\- m'v\ that is the form of the influence of m
and V, has through some experience or other been pre-
viously suggested to us as the determinative and de-
cisive factor. If we renounce the use of the Newtonian
principles, then some o'ther specific experiences con-
cerning the import oi mv which are equivalent to those
principles, are indispensable.

7. The impact of elastic masses may also be treated
by the Newtonian principles. The sole observation
here required is, that a deformation of elastic bodies
calls into play forces of restitutioriy which directly de-
pend on the deformation. Furthermore, bodies pos-
sess impenetrability ; that is to say, when bodies af-
fected with unequal velocities meet in impact, forces
which equalise these velocities are produced. If two
elastic masses My m with the velocities C, c collide, a
deformation will be effected, and this deformation will
not cease until the velocities of the two bodies are
equalised. At this instant, inasmuch as only internal
forces are involved and therefore the momentum and

Second, the
impact of
elastic
masses in
Newton's
view.

THE EXTENSION OF THE PRINCIPLES. 321

the motion of the centre of gravity of the system re-
main unchanged, the common equalised velocity will be

MC -\- mc
M -\- m '

Consequently, up to this time, Af^s velocity has suf-
fered a diminution C — u\ and nCs an increase u — c.
But elastic bodies being bodies that recover their
forms, in perfectly elastic bodies the very same forces
that produced the deformation, will, only in the in-
verse order, again be brought into play, through the
very same elements of time and space. Consequently,
on the supposition that m is overtaken by J/, M will a
second time sustain a diminution of velocity C — «, and
m will a second time receive an increase of velocity
u — c. Hence, we obtain for the velocities V, v after
impact the expressions Vz=^ 2u — C and v = 2u — r, or

AfC+m(2e—C) _ mc + M(2C—c)
If in these formulae we put J/ =w, it will follow T*>« ?«*«?-

'^ ' uon by this

that V= c and t/ = C ; or, if the impinging masses are ^'^ <>' »»
equal, the velocities which they have will be inter-
changed. Again, since in the particular case Mjm =
— cjC or MC -\- mc:=i^ also « = 0, it follows that
V=i2u— C=— C and v=%u — € = — c\ that is,
the masses recede from each other in this case with the
same velocities (only oppositely directed) with which
they approached. The approach of any two masses
J/, m affected with the velocities C, r, estimated as
positive when in the same direction, takes place with
the velocity C — c\ their separation with the velocity
V — V, But it follows at once from F=2u — C,
vt=2u — Cy that V — v = — (C — c); that is, the rela-
tive velocity of approach and recession is the same.

view.

322 THE SCIENCE OF MECHANICS.

By the use of the expressions F= 2i^ — Cand v =
2 « — r, we also very readily find the two theorems

MV -{- mv = MC -\- mc and

which assert that the quantity of motion before and
after impact, estimated in the same direction, is the
same, and that also the 7ns viva of the system before
and after impact is the same. We have reached, thus,
by the use of the Newtonian principles, all of Huy-
gens's results.
The impii- 8. If we consider the laws of impact from Huygens's
Huygens'8 point of view, the following reflections immediately
claim our attention. The height of ascent which the
centre of gravity of any system of masses can reach is
given by its vis viva, ^2mv^. In every case in which
work is done by forces, and in such cases the masses
follow the forces, this sum is increased by an amount
equal to the work done. On the other hand, in every
case in which the system moves in opposition to forces,
that is, when work, as we may say, is donf upon the
system, this sum is diminished by the amount of work
done. As long, therefore, as the algebraical sum of
the work done on the system and the work done by the
system is not changed, whatever other alterations may
take place, the sum^^/w?'^ also remains unchanged.
Huygens now, observing that this first property of ma-
terial systems, discovered by him in his investigations
on the pendulum, also obtained in the case of impact,
could not help remarking that also the sum of the
vires viva must be the. same before and after im-
pact. For in the mutually effected alteration of the
forms of the colliding bodies the material system con-
sidered has the same amount of work done on it as, on

THE EXTENSION OF THE PRINCIPLES. 323

the reversal of the alteration, is done by it, provided al-
ways the bodies develop forces wholly determined by
the shapes they assume, and that they regain their
original form by mean& of the same forces employed to
effect its alteration. That the latter process takes
place, definite experience alone can inform us. This law
obtains, furthermore, only in the case of so-called per-
fectly elastic bodies.

Contemplated from this point of view, the majority The dedac-
of the Huygenian laws of impact follow at once. Equal laws of im-

. , , pact by the

masses, which strike each other with equal but oppo- notion of

vis viva and

site velocities, rebound with the same velocities. The work,
velocities are uniquely determined only when they are
equals and they conform to the principle of vis viva
only by being the same before and after impact. Fur-
ther it is evident, that if one of the unequal masses in
impact change only the sign and not the magnitude of
i ts velocity, this must also be the case with the other.
On this supposition, however, the relative velocity of
separation after impact is the same as the velocity of
approach before impact. Every imaginable case can
be reduced to this one. Let c and c' be the velocities
of the mass m before and after impact, and let them be
of any value and have any sign. We imagine the whole
system to receive a velocity u of such magnitude that
« -f r = — {u-\- c'^ or u=l{c — ^')/2. It will be seen
thus that it is always possible to discover a velocity of
transportation for the system such that the velocity of
one of the masses will only change its sign. And so
the proposition concerning the velocities of approach
and recession holds generally good.

As Huygens's peculiar group of ideas was not fully
perfected, he was compelled, in cases in which the ve-
locity-ratios of the impinging masses were not origin-

324

THE SCIENCE OF MECHANICS,

Hnygens's
tacit appro-
priation of
the idea of
mass.

Construc-
tive com-
parison of
the special
and general
case of im-
pact.

ally known, to draw on the Galileo-Newtonian system
for certain conceptions, as was pointed out above.
Such an appropriation of the concepts mass and mo-
mentum, is contained, although not explicitly ex-
pressed, in the proposition according to which the ve-
locity of each impinging mass simply changes its sign
when before impact MIm = — c/C, If Huygens had
wholly restricted himself to his own point of view, he
would scarcely have discovered this proposition, al-
though, once discovered, he was able, after his own
fashion, to supply its deduction. Here, owing to the
fact that the momenta produced are equal and oppo-
site, the equalised velocity of the masses on the com-
pletion of the change of form will be » = 0. When the
alteration of form is reversed, and the same amount of
work is performed that the system originally suffered,
the satfie velocities with opposite signs will be restored.
If we imagine the entire system affected with a ve-
locity of translation^ this particular case will simulta-
neously present the^^«^rd:/case.
Let the impii^ging masses be
represented in the figure by
M—BC and m = AC (Fig.
164), and their respective velo-
cities by C= AD and c = BE.
On AB erect the perpendicular
CF, and through F draw IK
parallel to AB. Then ID = {m. C—c)/{,M-\- m) and
KB = {M.C — c)/{^M-\- m). On the supposition now
that we make the masses M and m collide with the
velocities ID and KEj while we simultaneously impart
to the system as a whole the velocity

u = AI= KB = C— {m . C—'~c)/{M-\- m) =
c + {M. C—c)J(^M-\- m) = {MC+ mc)/(^M+ m),

Fig. 164.

THE EXTENSION^ OF THE PRINCIPLES. 325

the spectator who is moving forwards with the velocity
u will see the particular case presented, and the spec-
tator who is at rest will see the general case, be the
velocities- what they may. The general formulae of im-
pact, above deduced, follow at once from this concep-
tion. We obtain :

M -^ m M -\- m

M -\- m M -\- m

Huygen*s successful employment of the fictitious signifi-
motions is the outcome of the simple perception that fictitious

, . motions.

bodies not affected with differences of velocities do not
act on one another in impact. All forces of impact are
determined by differences of velocity (as all thermal
effects are determined by differences of temperature).
And since forces generally determine, not velocities,
but only changes of velocities, or, again, differences of
velocities, consequently, in every aspect of impact the
sole decisive factor is differences of velocity. With re-
spect to which bodies the velocities are estimated, is
indifferent. In fact, many cases of impact which from
lack of practice appear to us as different cases, turn
out on close examination to be one and the same.

Similarly, the capacity of a moving body for work, velocity, a
whether we measure it with respect to the time of its level,
action by its momentum or with respect to the distance
through which it acts by its vis viva, has no signifi-
cance referred to a single body. It is invested with
such, only when a second body is introduced, and, in
the first case, then, it is the difference of the veloci-
ties, and in the second the square of the difference that
is decisive. Velocity is a physical level, like tempera-
ture, potential function, and the like.

326 THE SCIENCE OF MECHANICS.

Possible It remains to be remarked, that Huygens could

origin of have reached, originally, in the investigation of the

Huygens's , , . . i , ,

ideas. phenomena of impact, the same results that he pre-
viously reached by his investigations of the pendulum.
In every case there is one thing and one thing only to
be done, and that is, to discover in all the facts the same
elements^ or, if we will, to rediscover in one fact the
elements of another which we already know. From
which facts the investigation starts, is, however, a
matter of historical accident.
Conserva- 9. Let US close our examination of this part of the
mentum in- subject with a few general remarks. The sum of the
momenta of a system of moving bodies is preserved in
impact, both in the case of inelastic and elastic bodies.
But this preservation does not take place precisely in
the sense of Descartes. The momentum of a body is
not diminished in proportion as that of another is in-
creased ; a fact which Huygens was the first to note.
If, for example, two equal inelastic masses, possessed
of equal and opposite velocities, meet in impact, the
two bodies lose in the Cartesian sense their entire mo-
mentum. If, however, we reckon all velocities in a
given direction as positive, and all in the opposite as
negative, the sum of the momenta is preserved. Quan-
tity of motion, conceived in this sense, is always pre-
served.

The vis viva of a system of inelastic masses is al-
tered in impact \ that of a system of perfectly elastic
masses is preserved. The diminution of vis viva pro-
duced in the impact of inelastic masses, or produced
generally when the impinging bodies move with a com-
mon velocity, after impact, is easily determined. Let
My m be the masses, C, c their respective velocities be-

THE EXTENSION OF THE PRINCIPLES. 327

fore impact, and u their common velocity after impact ; Consenra-

- . . . tion of vis

then the loss of VtS viva is viva in im-

pact inter-

iJ/^C2 + Jw^a — i(ilf +»l)«a, (l)Preted.

which in view of the fact that u = {MC -\- tn c)/{M-\- tn)
may be expressed in the form {Mm/M -{- m)(^C — r)*.
Carnot has put this loss in the form

iM(C—uy-\-im(^u^cy (2)

If we select the latter form, the expressions ^M^C — u)^
and im{u — r)^ will be recognised as the vis viva gen-
erated by the work of the internal forces. The loss of
vis viva in impact is equivalent, therefore, to the work
done by the internal or so-called molecular forces. If
we equate the two expressions (i) and (2), remember-
ing that {M -|- w) « =: MC -^ me, we shall obtain an
identical equation. Carnot's expression is important
for the estimation of losses due to the impact of parts
of machines.

In all the preceding expositions we have treated pbUque
the impinging masses as points which moved only in the
direction of the lines joining them. This simplifica-
tion is admissible when the centres of gravity and the
point of contact of the impinging masses lie in one
straight line, that is, in the case of so-called direct im-
pact. The investigation of what is called oblique im-
pact is somewhat more complicated, but presents no
especial interest in point of principle.

A question of a different character was treated by The centre
Wallis. If a body rotate about an axis and its motion aioS!*^*^"*^
be suddenly checked by the retention of one of its
points, the force of the percussion will vary with the
position (the distance from the axis) of the point ar-
rested. The point at which the intensity of the impact
is greatest is called by Wallis the centre of percussion.

328 THE SCIENCE OF MECHANICS.

If this point be checked, the axis will sustain no pres-
sure. We have no occasion here to enter in detail
into these investigations ; they were extended and de-
veloped by Wallis's contemporaries and successors in
many ways.
Thebaiiis- lo. We will uow briefly examine, before concluding

tic pendu- - . ,

lam. this section, an interesting application of the laws of

impact ; namely, the determination of the velocities of
projectiles by the ballistic pendulum, A mass M\s sus-
pended by a weightless and massless
string (Fig. 165), so as to oscillate as a
pendulum. While in the position of
equilibrium it suddenly receives the hori-
zontal velocity V, It ascends by virtue
of this velocity to an altitude // = (/)
(1 — cos a) = V^ /2g, where /denotes the
length of the pendulum, a the angle of
elongation, and g the acceleration of
gravity. As the relation 7'= nV Ijg subsists between
the time of oscillation T and the quantities /, g^ we
easily obtain F= (^7y;r)l/2(l — cosa), and by the
use of a familiar trigonometrical formula, also

r r 2 or

V=-gTsm-^^,

Its formula. If now the velocity Fis produced by a projectile of

the mass m which being hurled with a velocity v and

sinking in M is arrested in its progress, so that whether

the impact is elastic or inelastic, in any case the two

masses acquire after impact the r^w»i<7« velocity K, it

follows that mv=^ (^-\- ^) ^\ or» if ^ be sufficiently

small compared with M, also v = (^M/m)V\ whence

finally

2 M -, a

THE EXTENSION OF THE PRINCIPLES. 329

If it is not permissible to regard the ballistic pen- A different

deduction.

dulum as a simple pendulum, our reasoning, in con-
formity with principles before employed, will take the
following shape. The projectile m with the velocity v
has the momentum mv^ which is diminished by the
pressure/ due to impact in a very short interval of
time r to tnV. Here, then, tn{v — F) =/t, or, if V
compared with v is very small, mv =pr. With Pon-
celet, we reject the assumption of anything like in-
stantaneous forces, which generate instanter velocities.
There are no instantaneous forces. What has been
called such are very great forces that produce per-
ceptible velocities in very short intervals of time, but
which in other respects do not differ from forces that
act continuously. If the force active in impact cannot
be regarded as constant during its entire period of ac-
tion, we have only to put in the place of the expression
pr the expression Cpdt, In other respects the reason-
ing is the same.

A force equal to that which destroys the momentum The vis

viva ftnd

of the projectile, acts in reaction on the pendulum. If work of the

pendulum.

we take the line of projection of the shot, and conse-
quently also the line of the force, perpendicular to the
axis of the pendulum and at the distance b from it, the
moment of this force will be bp^ the angular accelera-
tion generated bp/^mr^, and the angular velocity pro-
duced in time r

b . pT bmv

The m's viva which the pendulum has at the end of
time r is therefore

b^m^v^

iip^2mr^ = i ^

mr^

330 THE SCIENCE OF MECHANICS.

The result, By virtuc of this vis viva the pendulum performs
the excursion a, and its weight Mg^ {a being the dis-
tance of the centre of gravity from the axis,) is lifted
the distance a(\ — cos^f). The work performed here
is Mga{\ — cosflf), which is equal to the above-men-
tioned vis viva. Equating the two expressions we
readily obtain

1/2 Mga 2m r^ (1 — cos a)

mb '

and remembering that the time of oscillation is

and employing the trigonometrical reduction which
was resorted to immediately above, also

2 M a _ , a

V =. -.- gT , sm- .

n m b 2

interpreta- This formula is in every respect similar to that ob-

tion of the '' '^

result. tained for the simple case. The observations requisite
for the determination of v, are the mass of the pendu-
lum and the mass of the projectile, the distances of
the centre of gravity and point of percussion from the
axis, and the time and extent of oscillation. The form-
ula also clearly exhibits the dimensions of a velocity.
The expressions 2/;r and sin (a/2) are simple num-
bers, as are also M/m and a/b, where both numerators
and denominators are expressed in units of the same
kind. But the factor ^T' has the dimensions //"^, and
is consequently a velocity. The ballistic pendulum
was invented by Robins and described by him at length
in a treatise entitled JV^w Principles of Gunnery, pub-
lished in 1742.

THE EXTENSION OF THE PRINCIPLES. 331

d'alembert's principle.

1. One of the most important principles for the History of
rapid and convenient solution of the problems of me- cipie.
chanics is the principle of D^ A Umber i. The researches
concerning the centre of oscillation on which almost all
prominent contemporaries and successors of Huygens

had employed themselves, led directly to a series of
simple observations which D'Alembert ultimately gen-
eralised and embodied in the principle which goes by
his name. We will first cast a glance at these prelim-
inary performances. They were almost without excep-
tion evoked by the desire to replace the deduction of
Huygens, which did not appear sufficiently obvious, by
one that was more convincing. Although this desire was
founded, as we have already seen, on a miscompre-
hension due to historical circumstances, we have, of
course, no occasion to regret the new points of view
which were thus reached.

2. The first in importance of the founders of the James Ber-

noulli's

theory of the centre of oscillation, after Huygens, iscomribu-

tions to the

Tames Bernoulli, who sought as early as 1686 to ex- theory of

, the centre

plain the compound pendulum by the lever. He ar- of osdiu-
rived, however, at results which not only were obscure
but also were at variance with the conceptions of Huy-
gens. The errors of Bernoulli were animadverted on
by the Marquis de L*Hopital in \h& Journal de Rotter-
dam^ in 1690. The consideration of velocities acquired
in infinitely small intervals of time in place of velocities
acquired m finite times — a consideration which the last-
named mathematician suggested — led to the removal

332 THE SCIENCE OF MECHANICS.

of the main difficulties that beset this problem ; and in
1691, in the Acta ErudiUrum, and, later, in 1703, in the
Proceedings of the Paris Academy James Bernoulli cor-
rected his error and presented his results in a final and
complete form. We shall here reproduce the essential
points of his final deduction.
James Ber- A horizontal, massless bar AB (Fig. 166) is free to

noulli'sde- >^ v^ y

duction of rotate about A : and at the distances r, r' from A the

the law of . . - . -

the com- masses m. m* are attached. The accelerations with which

pound pen-

duiumfrom these masses as thus connected

the princi- nUr' X fikr

pie of the g^ '^ — [^>-^A ^^^^ ^^^^ must be different from

the accelerations which they
„. ^^ would assume if their connec-

FlR. 166.

tions were severed and they fell
freely. There will be one point and one only, at the
distance jc, as yet unknown, from A which will fall
with the same acceleration as it would have if it were
free, that is, with the acceleration g. This point is
termed the centre of oscillation.

\i m and ;«' were to be attracted to the earth, not
proportionally to their masses, but m so as to fall when
free with the acceleration cp = grjx and m* with the
acceleration <p' = gr^/x, that is to say, if the natural
accelerations of the masses were proportional to their
distances from A^ these masses would not interfere with
one another when connected. In reality, however, m
sustains, in consequence of the connection, an upward
component acceleration g — ^, and ;//' receives in virtue
of the same fact a downward component acceleration
cp' — g\ that is to say, the former suffers an upward
force of nt(^g — 9?) = g{x — r/a*) m and the latter a
downward force of m' i^cp' — g) == g{P — x/x) m\

Since, however, the masses exert what influence
they have on each other solely through the medium of

THE EXTENSION OF THE PRINCIPLES. 333

the lever by which they are joined, the upward force The lawof
upon the one and the downward force upon the other bution of

the effects

must satisfy the law of the lever. If m in conse- of the im-
quence of its being connected with the lever is held forces, in
back by a force /from the motion which it would take, nouiii's ex-
if free, it will also exert jthe same force /on the lever-
arm r by reaction. It is this reaction pull alone that
can be transferred to m and be balanced there by a
pressure /*= {r/r')/, and is therefore equivalent to the
latter pressure. There subsists, therefore, agreeably
to what has been above said, the relation g(y — -^A)
///' = r// * g{x — r/x) m or, {x — r)mr = (y — x) m'r^,
from which we obtain x=^{mr^ + m'/^)/{mr -[- wV),
exactly as Huygens found it. The generalisation of
this reasoning, for any number of masses, which need
not lie in a single straight line, is obvious.

3. John Bernoulli (in 1712) attacked in a different The prin-
manner the problem of the centre of oscillation. Hisjc£nBer-
performances are easiest consulted in his Collected inUon of
Works iOperay Lausanne and Geneva, 1762, Vols. IIof^tEecen"
and IV). We shall examine in detail here the main lation.
ideas of this physicist. Bernoulli reaches his goal by
conceiving the masses ^n^ forces separated.

First y let us consider two simple pendulums of dif- The first
ferent lengths /, /' whose bobs are affected with gravi- BernouiH"
tational accelerations proportional to the lengths of the
pendulums, that is, let us put ///' z=ig/g\ As the time
of oscillation of a pendulum is 2"= nV l/g, it follows
that the times of oscillation of these pendulums will be
the same. Doubling the length of a pendulum, ac-
cordingly, while at the same time doubling the accel-
eration of gravity does not alter the period of oscilla-
tion.

Second, though we cannot directly alter the accel-

334 THE SCIENCE OF MECHANICS.

The second eration of gravity at any one spot on the earth, we

step in John . ,i i • r«i • •

Bernoulli's can do what amounts virtually to this. Thus, imagine
a straight massless bar of length 2a, free to rotate about
its middle point; and attach to the one ex-
m' tremity of it the mass m and to the other the
a mass m\ Then the total mass is m -(- m* at
the distance a from the axis. But the force
« which acts on it is {m — m'^ gy and the ac-
m celeration, consequently, {m — tn' jm + w') g.
Fig. 167. Hence, to find the length of the simple pen-
dulum, having the ordinary acceleration of
gravity gy which is isochronous with the present pen-
dulum of the length «, we put, employing the preced-
• ing theorem,

/ g m-{- m

= , ox l = a - - -,

a tn — ft I m — m

m -\- m'^

The third Thirds we imagine a simple pendulum of length i

step, or the . , , . n-»i

detennina- With the mass tn at its extremity. The weight of tn
centre of produces, by the principle of the lever, the same ac-
celeration as half this force at a distance 2 from the
point of suspension. Half the mass m placed at the
distance 2, therefore, would suffer by the action of the
force impressed at i the same acceleration, and a fourth
of the mass m would suffer double the acceleration ; so
that a simple pendulum of the length 2 having the orig-
inal force at distance i from the point of suspension
and one-fourth the original mass at its extremity would
be isochronous with the original one. Generalising
this reasoning, it is evident that we may transfer any
force / acting on a compound pendulum at any dis-
tance r, to the distance i by making its value r/, and
any and every mass placed at the distance r to the
distance i by making its value r^m^ without changing

gyration.

THE EXTENSION OF THE PRINCIPLES. 335

the time of oscillation of the pendulum. If a force /
act on a lever-arm a (Fig. i$8) while at the distance r
from the axis a mass m is attached, / will be equiva-
lent to a force afjr impressed on
m and will impart to it the linear ^ ^

acceleration a//m r and the angu- ^

lar acceleration aflmr^. Hence,

to find the angular acceleration

of a compound pendulum, we ^^' '

divide the sum of the- statical moments by the sum of

the moments of inertia. . . n ''/ /

Brook Taylor, an Englishman,* also developed The re-

CAar^nAc of

this idea, on substantially the same principles, but Brook Tay
quite independently of John Bernoulli. His solution,
however, was not published until some time later, in
1 7 15, in his work, Methodus Increment or um.

The above are the most important attempts to solve
the problem of the centre of oscillation. We shall see
that they contain the very same ideas that D'Alembert
enunciated in a generalised form.

4. On a system of points J/", M\ M*\ . . . connected Motion of a
with one another in any way,t the forces P, F^ F", . . . pSmTsub-
are impressed. (Fig. 169.) These forces would im-strainto.
part to Xh^free points of the system certain determinate
motions. To the connected points, however, different
motions are usually imparted — motions which could
be produced by the forces IV, W, IV*\ , . , These
last are the motions which we shall study.

Conceive the force F resolved into W and V, the
force F' into W and V\ and the force F" into W"

* Author of Taylor's theorem, and also of a remarkable work on perspec-
tive.—Traiw.

t In precise technical language, they are subject to constraints, that is.
forces regarded as infinite, which compel a certain relation between their
motions. — TrAn*.

FiR. 169.

336 THE SCIENCE OF MECHANICS.

Statement and V*\ aod SO on. Since, owing to the connections,
berfs prin- Only the Components W^ W\ W*\ . . . are effective,

ciplc*

therefore, the forces K, V, V'\ . . . must be equilib-
rated by the connections. We will call the forces P, P\

P" the impressed forces,

the forces M^, ^, W^" ,

which produce the ac-
tual motions, the effective
forces, and the forces V,
V\ F" . . . . the forces
gained and lost, or the
equilibrated forces. We
perceive, thus, that if we
resolve the impressed forces into the effective forces
and the equilibrated forces, the latter form a system
balanced by the connections. This is the principle of
D'Alembert. We have allowed ourselves, in its expo-
sition, only the unessential modification of putting
forces for the momenta generated by the forces. In this
form the principle was stated by D'Alembert in his
Traits de dynamique J published in 1743.
Various As the system K, V\ V*\ ... is in equilibrium, the

forms in . . , / .

which the principle of Virtual displacements is applicable thereto.

may be ex- This gives a secoud form of D'Alembert*s principle.

A third form is obtained as follows : The forces P, P*

are the resultants of the components W, W\ . . . and
F, F'. . . . If, therefore, we combine with the forces

W, W and V, V the forces — P, —P' ,

equilibrium will obtain. The force-system — P, W^ V
is in equilibrium. But the system V is independently
in equilibrium. Therefore, also the system — /*, W\%
in equilibrium, or, what is the same thing, the system
Py — W^ is in equilibrium. Accordingly, if the effective
forces with opposite signs be joined to the impressed

THE EXTENSION OF THE PRINCIPLES.

337

Fig. vjo.

forces, the two, owing to the connections, will balance.
The principle of virtual displacements may also be ap-
plied to the system Py — W, This Lagrange did in his
Mecanique analytique^ 1788.

The fact that equilibrium subsists between the sys-
tem P and the system — W^ may be expressed in still
another way. We may say that
the system ^is equivalent to the _p
system P. In this form Her- '
MANN (Phoronotniay 1716) and
EuLER {Comment, Acad, Petrop,y
Old Series, Vol. VII, 1740) employed the principle.
It is substantially not different from that of D* Alembert.

5. We will now illustrate D'Alembert's principle by
one or two examples.

On a massless wheel and axle with the radii R^ r the
loads P and Q are hung, which are not in equilibrium.
We resolve the force P into (i) W
(the force which would produce the
actual motion of the mass if this were
free) and (2) F, that is, we put
P= W-\r ^and also Q = IV' + F';
it being evident that we may here
neglect all motions that are not in
the perponoiculwf. We have, accord-
ingly, V=P—W and F'= Q — W\ ^''^' '''"
and, since the forces F, V are in equilibrium, also
V,R= V\ r. Substituting for V, V in the last equa-
tion their values in the former, we get

An equiva-
lent princi-
ple em-
Rloyed by
[ermann
and Euler.

Illustration
of D'AIem-
bert'8 prin-
ciple by the
motion of a
wheel and
axle.

{P— IV)J{ = {Q— W')r

(1)

which may also be directly obtained by the employ-
ment of the second form of D'Alembert's principle.
From the conditions of the problem we readily perceive

338 THE SCIENCE OF MECHANICS.

that we have here to deal with a uniformly accelerated
motion, and that all that is therefore necessary is to
ascertain the acceleration. Adopting gravitation meas-
ure, we have the forces W and W\ which produce in
the masses Pjg and ^/f the accelerations y ^^^ y''>
wherefore, lV={P/g)y and W={Q/g)y\ But we
also know that y*= — yirjR). Accordingly, equation
(i) passes into the form

whence the values of the two accelerations are ob-
tained

PR—Qr. , , PR—Qr

^ PR^ + Qr^ "" ^ PR^ + Qr^

These last determine the motion.
Employ- It will be Seen at a glance that the same result can

ideas stat- be obtained by the employment of the ideas of statical
ment and moment and moment of inertia. We get by this method

moment of , . .

inertia, to for the angular acceleration

obtain this

'«*^*- _ PR—Qr __^Rj^Qr

g g

and as y = R<p and y'= — np^fe, re-obtain the pre-
ceding expressions.

When the masses and forces are given, the problem
of finding the motion of a system is determinate. Sup-
pose, however, only the acceleration y is given with
which P moves, and that the problem is to find the loads
P and Q that produce this acceleration. We obtain
easily from equation (2) the result P= Q{Rg-\- ry)
r/{g — y)R^, that is, a relation between /'and Q.
One of the two loads therefore is arbitrary. The prob-

THE EXTENSION OF THE PRINCIPLES.

339

lem in this form is an indeterminate one, and may be
solved in an infinite number of different ways.

The following may serve as a second example.

A weight P (Fig. 1 72) free to Inove on a vertical a second n
straight line ABy is attached to a cord

CX A

©x

passing over a pulley and carrying a
weight Q at the other end. The cord
makes with the line AB the variable
angle or. The motion of the present
case cannot be uniformly accelerated.
But if we consider only vertical mo-
tions we can easily give for every
value of a the momentary accelera-
tion {y and ^ ') oi P and Q. Proceeding exactly as
we did in the last case, we obtain

P= lV-\- F,

r .

Fig. 172,

of the prin-
ciple.

also

F' cos a = F, or, since y' = — ^ cos or,

Q \ P

Q + cos ar^ 1 cosor = P — y; whence

6 / o

P — Q cos a

y = —

Qcos^a-^-P
P — ^ cos or

cos a g.

Qcos^a + P

Again the same result may be easily reached by the solution of
employment of the Ideas of statical moment and mo-aisoby^Sie
ment of inertia in a more generalised form. The fol- statical mo-
lowing reflexion will render this clear. The force, or moment of
statical moment, that acts on P is P — Q cos a. But eraiiiSdf*'"
the weight Q moves cos a times as fast as P; conse-
quently its mass is to be taken cos ^a times. The ac-
celeration which P receives, accordingly is.

340 THE SCIENCE OF MECHANICS.

P — ^ COS or P — ^cosa

^"' ^ \ ~P^ 'Qco^^cr-\-~P ^'

g g

In like manner the corresponding expression for y may

be found.

The foregoing procedure rests on the simple re-
mark, that not the circular path of the motion of the
masses is of consequence, but only the relative veloci-
ties or relative displacements. This extension of the
concept moment of inertia may often be employed to
advantage.
Import and 6. Now that the application of D*Alembert*s prin-

character . * , . . ,

of D'Aiem- ciplc has been sufficiently illustrated, it will not be diffi-

bert'sprizi- '^ . "^ . '

cipie. cult to obtain a clear idea of its significance. Problems
relating to the motion of connected points are here dis-
posed of by recourse to experiences concerning the
mutual actions of connected bodies reached in the in-
vestigation of problems of equilibrium. Where the last
mentioned experiences do not su£Bce, D*Alembert*s
principle also can accomplish nothing, as the examples
adduced will amply indicate. We should, therefore,
carefully avoid the notion that D'Alembert's principle
is a general one which renders special experiences su-
perfluous. Its conciseness and apparent simplicity are
wholly due to the fact that it refers us to experiences
already in our possession. Detailed knowledge of the
subject under consideration founded on exact and mi-
nute experience, cannot be dispensed with. This knowl-
edge we must obtain either from the case presented,
by a direct investigation, or we must previously have
obtained it, in the investigation of some other subject,
and carry it with us to the problem in hand. We learn,
in fact, from D'Alembert's principle, as our examples
show, nothing that we could not also have learned by

THE EXTENSION OF THE PRINCIPLES. 341

Other methods. The principle fulfils in the solution
of problems, the office of a routine-form which, to a
certain extent, spares us the trouble of thinking out
each new case, by supplying directions for the employ-
ment of experiences before known and familiar to us.
The principle does not so much promote our insight
into the processes as it secures us 2i practical mastery of
them. The value of the principle is of an economical
character.

When we have solved a problem by D'Alembert's The reia-

^ ^ . tion of

principle, we may rest satisfied with the experiences D'Aiem-
previously made concerning equilibrium, the applica- cipie to the
tion of which the principle implies. But if we wish cipiea of

mechanics.

clearly and thoroughly to apprehend the phenomenon,
that is, to rediscover in it the simplest mechanical ele-
ments with which we are familiar, we are obliged to
push our researches further, and to replace our expe-
riences concerning equilibrium either by the Newtonian
or by the Huygenian conceptions, in some way similar
to that pursued on page 266. If we adopt the former
alternative, we shall mentally see the accelerated mo-
tions enacted which the mutual action of bodies on one
another produces ; if we adopt the second, we shall di-
rectly contemplate the work done, on which, in the
Huygenian conception, the vis viva depends. The latter
point of view is particularly convenient if we employ
the principle of virtual displacements to express the
conditions of equilibrium of the system V ox P — W.
D'Alembert*s principle then asserts, that the sum of
the virtual moments of the system V^ or of the system
P — Wy is equal to zero. The elementary work of the
equilibrated forces, if we leave out of account the strain-
ing of the connections, is equal to zero. The total
work done, then, is performed solely by the system /*,

342 THE SCIENCE OF MECHANICS.

and the work performed by the system ^must, accord-
ingly, be equal to the work done by the system P. All
the work that can possibly be done is due, neglecting
the strains of the connections, to the impressed forces.
As will be seen, D'Alembert's principle in this form is
not essentially difierent from the principle of vis viva.
Form of ap- 7. In practical applications of the principle of
D'Aiem- D'Alembcrt it is convenient to resolve every force P

bert'sprin- . , , , • . , .,

cipie, and impressed on a mass m of the system mto the mutually
inRequa- perpendicular components Xy V, Z parallel to the axes

tionS of mo- ' . . rr '

tion. of a system of rectangular coordinates ; every effective

force ^into corresponding components m^y mrfy m^,
where S, ?j, 5 denote accelerations in the directions of
the coordinates ; and every displacement, in a similar
manner, into three displacements Sx^ dy, dz. As the
work done by each component force is effective only in
displacements parallel to the directions in which the
components act, the equilibrium of the system (/*, — W)
is given by the equation

2\(^X^mS)Sx + (^V—mrf)dy+{Z—m^6z\ = 0 (1)

2(,X6x + VSy + ZSz) = 2m(S6x+f^6y+^6z). . (2)

These two equations are the direct expression of the
proposition above enunciated respecting the possible
work of the impressed forces. If this work be = 0, the
particular case of equilibrium results. The principle
of virtual displacements flows as a special case from
this expression of D'Alembert's principle ; and this is
quite in conformity with reason^ since in the general
as well as in the particular case the experimental per-
ception of the import of work is the sole thing of con-
sequence.

Equation (i) gives the requisite equations of mo-

THE RXTENSfON OF THE PRINCIPLES. 343

tion ; we have simply to express as many as possible
of the displacements Sx^ dy, Sz by the others in terms
of their relations to the latter, and put the coefficients
of the remaining arbitrary displacements = 0, as was
illustrated in our applications of the principle of vir-
tual displacements.

The solution of a very few problems by D'Alem- conve-

... •if/v* • "irii oicnce and

bert s pnnciple will sumce to impress us with a full atiiity of
sense of its convenience. It will also give us the con- bert's pnn-

ciplc.

viction that it is possible, in every case in which it may
be found necessary, to solve directly and with perfect
insight the very same problem by a consideration of
elementary mechanical processes, and to arrive thereby
at exactly the same results. Our conviction of the
feasibility of this operation renders the performance of
it, in cases in which purely practical ends are in view,
unnecessary*

VI.
THE PRINCIPLE OF VIS VIVA.

I. The principle of ins viva, as we know, was first The oiifc-
employed by Huvgens. John and Daniel Bernoulli icarform of
had simply to provide for a greater generality of ex- cipief"*

pression ; they added little. If /, /,/" are weights,

niy m\ m". . . . their respective masses, A, h\ K\ . . . the
distances of descent of the free or connected masses,
and r, v\ v". . . . the velocities acquired, the relation
obtains

If the initial velocities are not = 0, but are v^, v^,
v^\ . . ., the theorem will refer to the increment of the
7)is viva by the work and read

:Sph = :^:Sm[v^ — v^^),

344

THE SCIENCE OF MECHANICS,

Theprinci- The principle still remains applicable when / . . . .

pie applied • i «

to forces of are, not weights, but any constant forces, and A . . .
not the vertical spaces fallen through, but any paths in
the lines of the forces. If the forces considered are
variable, the expressions /^, /'^'. . . . must be replaced
by the expressions Cp ds^ Cp' ds* . . . . , in which / de-
notes the variable forces and ds the elements of dis-
tance described in the lines of the forces. Then

The princi-
ple illus-
trated hy
the motion
of a wheel
and axle.

^Jpds = ^2m(j)^ — rj^)

(1)

2. In illustration of the principle of vts viva we
shall first consider the simple problem which we treated

by the principle of D'Alembert. On
a wheel and axle with the radii R, r
hang the weights P^ Q. When this
machine is set in motion, work is per-
formed by which the acquired vis viva
is fully determined. For a rotation of
the machine through the angle or, the
work is

P,Ra—Q. ra = a{PR — Qr),
Calling the angular velocity which
corresponds to this angle of rotation, ^, the vis viva
generated will be

Consequently, the equation obtains

Fig. 173.

a(PR - Qr) = T^- {PJi^ + Qr^)

(1)

Now the motion of this case is a uniformly accelerated
motion ; consequently, the same relation obtains here
between the angle a, the angular velocity ^, and the

THE EXTENSION OF THE PRINCIPLES.

345

angular acceleration ^% as obtains in free descent be-
tween X, 7/, g. If in free descent s = v^ /2gy then here

Introducing this value of a in equation (i), we get
for the angular acceleration of P^ ^ = (-P^ — Qr/
m^ -j- Qr^)g, and, consequently, for its absolute ac-
celeration y = {FR^^QrlPR'^ + ~Qr'^) Rg, exactly as
in the previous treatment of the problem.

As a second example let us consider the case of a a rpiiinR

. , cylinder on

massless cylinder of radius r, in the surface of which, an inclined

plane.

diametrically opposite each other, are fixed two equal
masses m, and which in consequence of the weight of

Fig. 174.

these masses rolls without sliding down an inclined
plane of the elevation or. First, we must convince our-
selves, that in order to represent the total vis viva of
the system we have simply to sum up the vis viva of
the motions of rotation and progression. The axis of
the cylinder has acquired, we will say, the velocity u
in the direction of the length of the inclined plane, and
we will denote by v the absolute velocity of rotation of
the surface of the cylinder. The velocities of rotation v
of the two masses ni make with the velocity of progres-
sion u the angles B and d* (Fig. 175), where 6^-1- 6^'
= i8o^ The compound velocities w and z satisfy
therefore the equations

W' = «2 _j_ ^2 — 2uvcosB

«a =«2 + z;a — 2uvcosd',

346 THE SCIENCE OF MECHANICS.

The law of But since COS 6 = — COS S'y it follows that

motion of

If the cylinder moves through the angle <p, m describes
in consequence of the rotation the space r q>y and the
axis of the cylinder is likewise displaced a distance r<p.
As the spaces traversed are to each other, so also
are the velocities v and «, which therefore are equal.
The total vis viva may accordingly be expressed by
2«r//2. If /is the distance the cylinder travels along
the length of the inclined plane, the work done is
2»i^. /sinar = 2/^«2j whence « = v/^/. sin or. If we
compare with this result the velocity acquired by a body
in sliding down an inclined plane, namely, the velocity
V 7,gl sin^, it will be observed that the contrivance we
are here considering moves with only one-half the ac-
celeration of descent that (friction neglected) a sliding
body would under the same circumstances. The rea-
soning of this case is not altered if the mass be uni-
formly distributed over the entire surface of the cylin-
der. Similar considerations are applicable to the case
of a sphere rolling down an inclined plane. It will be
seen, therefore, that Galileo's experiment on falling
bodies is in need of a quantitative correction.
A modifica- Ncxt, let US distribute the mass m uniformly over
preceding the surface of a cylinder of radius R^ which is coaxal

case*

with and rigidly joined to a massless cylinder of radius
r, and let the latter roll down the inclined plane. Since
here vfu = R/r, the principle of vis viva gives mgl
sina = ^mu^(l + R^/r^), whence

2g/ sin a

THE EXTENSION OF THE PRINCIPLES. 347

For Rjr = i the acceleration of descent assumes its
previous value gji. For very large values of Rjr the
acceleration of descent is very small. When Rjr = 00
it will be impossible for the machine to roll down the
inclined plane at all.

As a third example, we will consider the case of a The motion

. . of a chain

chain, whose total length is /, and which lies partly on on an in-
a horizontal plane and partly on a plane having the plane,
angle of elevation a. If we imagine the surface on
which the chain
rests to be very
smooth, any very
small portion of
the chain left hang-

. . Fig. 176.

mg over on the m-

clined plane will draw the remainder after it. If /i is
the mass of unit of length of the chain and a portion x
is hanging over, the principle of vis viva will give for
the velocity v acquired the equation

ixlv'^ X . X

or V =^x\/ g ^v^oLJL In the present case, therefore,
the velocity acquired is proportional to the space de-
scribed. The very law holds that Galileo first con-
jectured was the law of freely falling bodies. The
same reflexions, accordingly, are admissible here as at
page 248.

3. Equation (i), the equation of vis viva, can always Extension
be employed, to solve problems of moving bodies, dpie%?"?i
when the total distance traversed and the force that*"*"*'
acts in each element of the distance are known. It was
disclosed, however, by the labors of Euler, Daniel Ber-
noulli, and Lagrange, that cases occur in which the

348 THE SCIENCE OF MECHANICS.

principle of vis viva can be employed without a knowl-
edge of the actual path of the motion. We shall see
later on that Clairaut also rendered Important services
in this field,
inen- Galileo, even, knew that the velocity of a heavy

Enier. falling body depended solely on the vertical height de-
scended through, and not on the length ox/orm of the
path traversed. Similarly, Hu)^ens finds that the vis
viva of a heavy material system is dependent on the
vertical heights of the masses of
the system. Euler was able to
make a further step in advance.
If a body K (Fig. 177) is at-
tracted towards a fixed centre
C in obedience to some given
law, the increase of the vis viva
in the case of rectilinear ap-
proach is calculable from the
initial and terminal distances
pj^ ,^_ (r^, r,). But the increase is the

same, if ^passes at all from the
position fg to the position r„ independently of the
form of its path, KB. For the elements of the work
done must be calculated from the projections on the
radius of the actual displacements, and are thus ulti-
mately the same as before.
Tiie te- UK is attracted towards several fixed centres C,

Daniel Ber- C, C". . , ,, the increase of its vis viva depends on the

noulUBnd ...,,. , „ , . - ,

Lagrange, initial distances r^, r^, r^ . . . . and on the terminal
distances r„ r,', r". . . ., that is on the initial and \x^t-
rciYRsX positions of K. Daniel Bernoulli extended this
idea, and showed further that where movable bodies
are in a state of mutual attraction the change of I'is viva
is determined solely by their initial and terminal dis-

THE EXTENSION OF THE PRINCIPLES. 349

tances from one another. The analytical treatment of
these problems was perfected by Lagrange. If we join
a point having the coordinates a, by c with a point hav-
ing the coordinates x^ y, z, and denote by r the length
of the line of junction and by a, /3, y the angles that
line makes with the axes of x, y, z, then, according to
Lagrange, because

ra = (j^ — «)» + Cr — by + {z — cY,

X — a dr - y — b dr
cosa = = -^ ,cosp= =-^f

r dx r dy

z — c dr

cos Y =ae =: — -.

' r dz

Accordingly, it/(r) = — -^-J- is the repulsive force, or The force

af compo-

nents, par-

the negative of the attractive force acting between the tiai diser-

. ential coef-

two points, the components will be fidentsof

the same

•^ ^ ^ dr dx dx "^^•«-

y=/(r)cos^ = -^-^^^=-^^-^

_ . , ^ * dF{r) dr dF{f)

Z =/{r) cosy = ,^^ -,- = -,- -.
"^ ^ ^ '^ dr dz dz

The force-components, therefore, are the partial'
difierential coefficients of one and the same function of
r, or of the coordinates of the repelling or attracting
points. Similarly, if several points are in mutual ac-
tion, the result will be

dx
dU
~ dy
^_dU
dz'

350 THE SCIENCE OF MECHANICS,

The force- wherc 6^ is a function of the coordinates of the points.
This function was subsequently called by Hamilton*
the force-function.

Transforming, by means of the conceptions here
reached, and under the suppositions given, equation
(i) into a form applicable to rectangular coordinates,
we obtain

^C{Xdx + Ydy + Zdz) = 2\m{v^ — v,^) or,

since the expression to the left is a complete differen-
tial,

2JdU=2(,U,— U:) = 2im(^v^ — v^^),

where C/^ is a function of the terminal values and ^
the same function of the initial values of the coordi-
nates. This equation has received extensive applica-
tions, but it simply expresses the knowledge that under
the conditions designated the work done and therefore
also the vis viva of a system is dependent on the posi-
tions^ or the coordinates, of the bodies constituting it.
If we imagine all masses fixed and only a single
one in motion, the work changes only as U changes.
The equation 6^= (constant defines a so-called level
surface, or surface of equal work. Movement upon
such a surface produces no work. U increases in the
direction in which the forces tend to move the bodies.

VII.

THE PRINCIPLE OF LEAST CONSTRAINT.

I. Gauss enunciated (in Cxe\\e*s Journal fUr Mathe^
matikj Vol. IV, 1829, p. 233) a new law of mechanics,
the principle of least constraint. He observes, that, in

* On a General Method in Dynamics, PhiL Trans, for 1834. See also C. G.
J. Jacobi, VorUsungen Uber Dynamik^ edited by Clebsch, 1866.

THE EXTENSION OF THE PRINCIPLES. 351

the form which mechanics has historically assumed, dy- History of
namics is founded upon statics, (for example, D'Alem- pie of least
bert's principle on the principle of virtual displace-
ments,) whereas one naturally would expect that in
the highest stage of the science statics would appear
as a particular case of dynamics. Now, the principle
which Gauss supplied, and which we shall discuss in
this section, includes both dynamical and statical cases.
It meets, therefore, the requirements of scientific and
logical aesthetics. We have already pointed out that this
is also true of D'Alembert's principle in its Lagrangian
form and the mode of expression above adopted.
No essentially new principle ^ Gauss remarks, can now be
established in mechanics ; but this does not exclude
the discovery of new points of view, from which mechan-
ical phenomena may be fruitfully contemplated. Such
a new point of view is afforded by the principle of
Gauss.

2. Let m, >»,.... be masses, connected in any man- statement
ner with one another. These masses, if/r^^, would, under cipie.
the action of the forces im-
pressed on them, describe in a
very short element of time the
spaces a b, a, bf , , . ,\ but in
consequence of their connec-
tions they describe in the same
element of time the spaces a r,

tff r/. . . . Now, Gauss's principle asserts, that the mo-
tion of the connected points is such that, for the motion
actually taken, the sum of the products of the mass of
each material particle into the square of the distance of
its deviation from the position it would have reached if
free, namely »i(^r)2 -|- m,\b'y,y -f . . . .= 2m{bcyy is
a minimum, that is, is smaller for the actual motion

352 THE SCIENCE OF MECHANICS,

than for any other conceivable motion in the same con-
flections. If this sum, 2m(Jfc)^f is less for rest than
for any motion, equilibrium will obtain. The principle
includes, thus, both statical and dynamical cases.
Definition The sum 2 m(fie)^ is called the "constraint.*** In

of "con- , ^ ,

straint." formmg this sum it is plain that the velocities present
in the system may be neglected, as the relative posi-
tions of a, d, c are not altered by them.

3. The new principle is equivalent to that of
D' Alembert ; it may be used in place of the latter ; and,
as Gauss has shown, can also be deduced from it. The
impressed forces carry the free mass m in an element of
time through the space a by the effective forces carry the
same mass in the same time in consequence of the con-
nections through the space a c. We resolve a b into a c

and cb\ and do the same for all the
masses. It is thus evident that
forces corresponding to the dis-
tances c bf Cfbf , , , , and propor-
tional thereto, do not, owing to the
connections, become effective, but
form with the connections an equilibrating system. If,
accordingly, we erect at the terminal positions c, c„
c,, .. . . the virtual displacements cy, c,y, . . . ,, form-
ing with cb, c,b, , , . . the angles 0, 0, , , . , we may
apply, since by D'Alembert's principle forces propor-
tional to cby c,bf , . , , are here in equilibrium, the
principle of virtual velocities. Doing so, we shall have

* Professor Mach's term is Abwetchun^ssumme. The AdweicAung- is the
declination or departure from free motion, called by Gauss the Ablenkung.
(See DQhring, Principien der Mechanik^ KS 168, 169 ; Routh, Rigid Dynamics^
Part I, II 390-394.) The quantity 2 m {bc)9 is called by Gauss the Zwang; and
German mathematicians usually follow this practice. In English, the term
constraint is established in this sense, although it is also used with another*
hardly quantitative meaning, for the force which restricts a body absolutely
to moving in a certain way. — Trans,

THE EX TENS 10 ^r OF THE PRINCIPLES. 353

"Zcb , CY CO%B^^ (l')Thc dednc-

'^ ^ ^ ^ tion of the

But principle

of least

{byy = {bey + {Cyy —2bc,CyCOSe, constraint,

IPyY — \bcy =(cyy — 2bc.cycose, and
2m{byy—2m{bcy=2m{cyy—22mbc.cycose(2)

Accordingly, since by (i) the second member of
the right-hand side of (2) can only be = 0 or negative,
that is to say, as the sum 2m(cyy can never be dimin-
ished by the subtraction, but only increased, therefore
the left-hand side of (2) must also always be positive
and consequently 2m(byy always greater than 2 m
{bey, which is to say, every conceivable constraint
from unhindered motion is greater than the constraint
for the actual motion.

4. The declination, be, for the very small element various

r • 1 1 forms in

of time r, may, for purposes of practical treatment, be which the
designated by j, and following Scheffler (Schlomilch's may be ex-
Zeitschrift fiir Mathematik und Physik, 1858, Vol. Ill,
p. 197), we may remark that s = yr^ ji, where y de-
notes acceleration. Consequently, 2ms^ may also be
expressed in the forms

where/ denotes the force that produces the declination
from free motion. As the constant factor in no wise
affects the minimum condition, we may say, the actual
motion is always such that

2ms^ (1)

2P^ (2)

2my^ (3)

is a minimum.

2m,s,s = ^^-2my,s= -2p.s= 2my^^

354

THE SCIENCE OF MECHANICS.

The motion
of a wheel
and axle.

5. We will first employ, in our illustrations, the
third form. Here again, as our first example, we se-
lect the motion of a wheel and axle by
the overweight of one of its parts
and shall use the designations above
frequently employed. Our problem
is, to so determine the actual accel-
erations y ol P and y, of Q, that

i^U) U - yy + ^Q/g) U - YfY

shall be a minimum, or, since y, =
— yir/R), so that P {g — yy +

Qis + Y'^/-^^ = ^ shall assume its smallest value.

Putting, to this end.

Fig. i8a

we get y = {^PR— Qr/PR^ + Qr^)Rg, exactly as in
the previous treatments of the problem.
Descent on As our secoud example, the motion of descent on
plane. ^^ inclined plane may be taken. In this case we shall

employ the first form, 2ms^.
Since we have here only to
deal with one mass, our in-
quiry will be directed to find-
ing that acceleration of de-
scent y for the plane by
which the square of the de-
clination (j^) is made a minimum. By Fig. 181 we
have

and putting d(^s^)/dy = 0, we obtain, omitting all
constant factors, 2y — 2^ sin or = 0 or y = g. sin or, the
familiar result of Galileo's researches.

Fig. x8x.

THE EXTENSION OF THE PRINCIPLES,

355

JiL

Fig. 182.

The following example will show that Gauss's prin- a case of
ciple also embraces cases of equilibrium. On the arms num.
a, a' of a lever (Fig. 182) are hung the heavy masses
m, m\ The principle requires that m(^g — y)^ -{■
m'(jr — y ')a shall be a minimum. But y'= — y(a'/a).
Further, if the masses are in-
versely ^ proportional to the
lengths of the lever-arms, that
is to say, if m/m' = a' /a, then
y' = — y (, m/m ' ). Conse-
quently, m{g — y) * + m\g + y • fft/m)^ = N must
be made a minimum. Putting dNjdy = 0, we get
m{i -\- m/m')y = 0 or y = 0. Accordingly, in this case
equiiibrium presents the least constraint from free mo-
tion.

Every new cause of constraint, or restriction upon New caasea
the freedom of motion, increases the quantity of con- straint in-

, , crease tho

Straint, but the increase is always the least possible, deparmre

from free

If two or more systems be connected, the motion of motion,
least constraint from the motions of the unconnected
systems is the actual motion.

If, for example, we join together several simple
pendulums so as to form a compound linear pendulum,
the latter will oscillate with the motion
of least constraint from the motion of the
single pendulums. The simple pendulum,
for any excursion a, receives, in the di-
rection of its path, the acceleration g
sin or. Denoting, therefore by y sin a the
acceleration corresponding to this excur-
sion at the axial distance i on the com-
pound pendulum, 2m (g sin a — rysina)^ or 2m (g —
ry)^ will be the quantity to be made a minimum. Conse-
quently, 2m(^g — ry)r = 0, and y = g(2mr/2mr^).

Fig. 183.

356

THE SCIENCE OF MECHANICS,

The problem is thus disposed of in the simplest man-
ner. But this simple solution is possible only because
the experiences that Huygens, the Bemoullis, and oth-
ers long before collected, are implicitly contained in
Gauss's principle,
iiiustra- 6. The increase of the quantity of constraint, or

preceding declination, from free motion by new causes of con-
straint may be exhibited by the following examples.

Over two stationary pulleys A^ B^ and beneath a
movable pulley C (Fig. 184), a cord is passed, each

TBwff?

Fig. 184.

extremity of which is weighted with a load P\ and on
C a load 7,P -f / is placed. The movable pulley will
now descend with the acceleration (J>/\P + /) g* But
if we make the pulley A fast, we impose upon the
system a new cause of constraint, and the quantity of
constraint, or declination, from free motion will be in-
creased. The load suspended from B, since it now
moves with double the velocity, must be reckoned as
possessing four times its original mass. The mova-
ble pulley accordingly sinks with the acceleration
{p/^P + p)g' A simple calculation will show that the
constraint in the latter case is greater than in the former.

THE EXTENSION OF THE PRINCIPLES,

357

A number, «, of equal weights, /, lying on a smooth
horizontal surface, are attached to n small movable
pulleys through which a cord is drawn in the manner
indicated in the figure and loaded at its free extremity
with /. According as all the pulleys are movable or all
except one zx^fixedy we obtain for the motive weight/,
allowing for the relative velocities of the masses as re-
ferred to /, respectively, the accelerations (4 «/i + 4«)^
and (4/5) g' If all the « + i masses are movable, the
deviation assumes the value/^/4« + i, which increases
as n, the number of the movable masses, is decreased.

w ^^

.^^

Fig. 186.

7. Imagine a body of weight Q, movable on rollers Treatment

of A me*

on a horizontal surface, and having an inclined plane chanicai

problein by

face. On this inclined face a body of weight P isdiflferent

. mechanical

placed. We now perceive instinctively that P will de- principles,
scend with quicker acceleration when Q is movable
and can give way, than it will when Q is fixed and /"s
descent more hindered. To any distance of descent h
of jP a horizontal velocity v and a vertical velocity u of
P and a horizontal velocity w ot Q correspond. Owing
to the. conservation of the quantity of horizontal mo-
tion, (for here only internal forces act,) we have Pv =
Qw, and for obvious geometrical reasons (Fig. 186)

also

u = (v -{- w) tan a

The velocities, consequently, are

u^=u

358 THE SCIENCE OF MECHANICS.

Q
First, by the 7;= ^ ^- -COtOT. »,

principles ■« "f' C:

of the con-
servation of p

viva. ■' ~r ^

For the work Fh performed, the principle of vis
viva gives

_^ Pu^ P( Q \^«^ .

2^2

cot

«)■

Multiplying by — , we obtain

J (^ , _Q cos^aN »a

To find the vertical acceleration y with which the
space h is described, be it noted that h=:u^ /2 y. In-
troducing this value in the last equation, we get

_(P+ 0sin»a

For Q = 00, y =g sin ^ a, the same as on a sta-
tionary inclined plane. For Q = 0, y =g, as in free
descent. For finite values of Q = mP,y/e get,

1 -\- PI ,

smce -.-_ — > 1,
sm^a -f- m

(1 + m)sin^a . „

The making of Q stationary, being a newly imposed
cause of constraint, accordingly increases the quantity
of constraint, or declination, from free motion.

To obtain y, in this case, we have employed the
principle of the conservation of momentum and the

THE EXTENSION OF THE PRINCIPLES. 359

principle of vis viva. Employing Gauss's principle, Second, by
we should proceed as follows. To the velocities de- cipie of

Gauss.

noted as », v^ w the accelerations y^ ^> ^ correspond.
Remarking that in the free state the only acceleration
is the vertical acceleration of jP, the others vanishing,
the procedure required is, to make

P P O

a minimum. As the problem possesses significance
only when the bodies P and Q touch, that is only when
y = ((J -j- £) tan or, therefore, also

Forming the differential coefficients of this expression
with respect to the two remaining independent vari-
ables 6 and f, and putting each equal to zero, we ob-
tain

_ \g—{d + €) tana] Ptantf + P6 = ^ and

From these two equations follows immediately
P6 — ^€ = 0, and, ultimately, the same value for y
that we obtained before.

We will now look at this problem from another
point of view. The body P describes at an angle §
with the horizon the space s, of which the horizontal
and vertical components are v and 1/, while simulta-
neously Q describes the horizontal distance w. The
force-component that acts in the direction of s is jPsin §y
consequently the acceleration in this direction, allow-
ing for the relative velocities of P and Q, is

/^.sin/?

36o THE SCIENCE OF MECHANICS,

Third, by Employing the following equations which are di-

tended con- rectly deducible,

cept of mo-

ment of in- Qw = Pv

ertia.

V =^ S COS fS

u=^v tan ft,

the acceleration in the direction of s becomes

Qsmft

and the vertical acceleration corresponding thereto is

^ '^'Q'+Pcos^' ^'

an expression, which as soon as we introduce by means
of the equation « = (z; -(- w) tan or, the angle-func-
tions of a for those of ft, again assumes the form above
given. By means of our extended conception of mo-
ment of inertia we reach, accordingly, the same result
as before.
Fourth, b^ Finally we will deal with this problem in a direct
cip?es.^"" manner. The body P does not descend on the mova-
ble inclined plane with the vertical acceleration g, with
which it would fall if free, but with a different vertical
acceleration, y. It sustains, therefore, a vertical coun-
terforce {P/g){g — y)- But as P and Q, friction
neglected, can only act on each other by means of a
pressure S, normal to the inclined plane, therefore

— {g — y) = Scosa and

0 P

5sin a = -- f == - 6,
g g

From this is obtained

-(g—y) = j€cota,

THE EXTENSION OF THE PRINCIPLES. 361

and by means of the equation ^ == (tf -[- 0 ^^^ ^» ^^^'
mately, as before,

(/'+ 0sin2a

PsvQ^a-^ Q

g (1)

^^sinacosor

Psm^a^ <2

P sin or cos or ,^

*'~?sin2a+ e^ ^ ^

If we put P ^=^ Q and or = 45°, we obtain for this Discusaion
*^ ^ ^•^ of the re-

particular case y = ^^, 6 = ^gf €z=^g. For P/g = suiu.

Q/g= I we find the ** constraint,* 'or declination from

free motion, to be ^ ^ 73. If we make the inclined plane

stationary, the constraint will be g^/2. If -Amoved on

a stationary inclined plane of elevation /3, where

tan /3 = y/d, that is to say, in the same path in which

it moves on the movable inclined plane, the constraint

would only be ^2/5. And, in that case it would, in

reality, be less impeded than if it attained the same

acceleration by the displacement of Q.

8. The examples treated will have convinced us that Gauss's

, . . principle

no substantially new insight or perception is afforded by affords no

new insisli^

Gauss's principle. ^ Employing form (3) of the prin-
ciple and resolving all the forces and accelerations in
the mutually perpendicular coordinate- directions, giv-
ing here the letters the same significations as in equa-
tion (i) on page 342, we get in place of the declination,
or constraint, 2my^, the expression

2-]

N=^m

— <?

(4)

and by virtue of the minimum condition

362 THE SCIENCE OF MECHANICS,

= 0.

or 2[(Ar— w ^ ^/^ + ( K— »i 7) //^ + (Z— OT <?y <2] = 0.
Gauss's and If no Connections exist, the coefficients of the (in

D'Alem- . ^

berfs prin- that case arbitrary) //^, dti. d^, severally made = 0,

ciples com- . , ,

mutable, give the equations of motion. But if connections do
exist, we have the same relations between dSt drfy dSi
as above in equation (i), at page 342, between rfjc, d}\
dz. The equations of motion come out the same ; as
the treatment of the same example by D'Alembert's
principle and by Gauss's principle fully demonstrates.
The first principle, however, gives the equations of
motion directly, the second only after differentiation.
If we seek an expression that shall give by differentia-
tion D'Alembert's equations, we are led perforce to the
principle of Gauss. The principle, therefore, is new
only in form and not in matter. Nor does it, further,
possess any advantage over the Lagrangian form of
D'Alembert's principle in respect of competency to com-
prehend both statical and dynamical problems, as has
been before pointed out (page 342).

The phys- There is no need of seeking a mystical or metaphys-

ical basis , . ,

of the prin- /Va/reason for Gauss's principle. The expression ' * least

ciple. ... . 1 . i. ,

constramt" may seem to promise somethmg of the
sort ; but the name proves nothing. The answer to the
question, *'/« what does this constraint consist ? " can-
not be derived from metaphysics, but must be sought
in the facts. The expression (2) of page 353, or (4) of
page 361, which is made a minimum, represents the
work done in an element of time by the deviation of the
constrained motion from the free motion. This work,
the work due to the constraint, is less for the motion
actually performed than for any other possible motion.

THE EXTENSION OF THE PRINCIPLES,

363

Once we have recognised work as the factor deter- Rdie of the

factor work.

minative of motion, once we have grasped the mean-
ing of the principle of virtual displacements to be, that
motion can never take place except where work can be
performed, the following converse truth also will in-
volve no difficulty, namely, that all the work that can
be performed in an element of time actually is per-
formed. Consequently, the total diminution of work
due in an element of time to the connections of the
system's parts is restricted to the portion annulled by
the counter-work of those parts. It is again merely a
new aspect of a familiar fact with which we have here
to deal.

This relation is displayed in the very simplest cases. The foun-

dations of

Let there be two masses m and m 2X A^ the one im- the princi-
ple reccg-

pressed with a force /, the other with nisabie m

the force q. If we connect the two, we . ,. ^0 piest cases,
shall have the mass 2 m acted on by a
resultant force r. Supposing the spaces
described in an element of time by the
free masses to be represented by A C,
A By the space described by the con-
joint, or double, mass will be AO =
^AD, The deviation, or constraint,
is m(^OB^ + OC^). It is less than
it would be if the mass arrived at the end of the ele-
ment of time in M or indeed in any point lying out-
side of B Cy say N, as the simplest geometrical con-
siderations will show. The deviation is proportional
to the expression p^ -\- q^ -\- '^pq cos ^/2, which in the
case of equal and opposite forces becomes 2/2^ and in
the case of equal and like-directed forces zero.

Two forces / and q act on the same mass. The
force q we resolve parallel and at right angles to the

Fig. 187.

364 THE SCIENCE OF MECHANICS,

Even in the direction of / in r and s. The work done in an element

principle of . .

the compo- of time is proportional to the squares of the forces, and

sition of , . . ., 1 «

forces its if there be no connections is expressible by /^ _|_ ^3 _-.

properties

are found, ^a -[- r ^ -(- j-^. If now r act directly counter to the
force /, a diminution of work will be effected and the
sum mentioned becomes (/ — r)^ + j^. Even in the
principle of the composition of forces, or of the mutual
independence of forces, the properties are contained
which Gauss's principle makes use of. This will best
be perceived by imagining all the accelerations simul-
taneously performed. If we discard the obscure verbal
form in which the principle is clothed, the metaphysical
impression which it gives also vanishes. We see the
simple fact ; we are disillusioned, but also enlightened.
The elucidations of Gauss's principle here presented
are in great part derived from the paper of SchefHer
cited above. Some of his opinions which I have been
unable to share I have modified. We cannot, for ex-
ample, accept as new the principle which he himself
propounds, for both in form and in import it is identical
with the D'Alembert-Lagrangian.

VIII.
THE PRINCIPLE OF LEAST ACTION.

Theorig- I. Maupertuis enunciated, in 1747, a principle

scurefortn which he Called ** le principe de la moindre quantity d'ac-

dpie*o?"" iion,^^ the principle of least action. He declared this

' principle to be one which eminently accorded with the

wisdom of the Creator. He took as the measure of

the "action" the product of the mass, the velocity,

and the space described, or m%)s, Why^ it must be

confessed, is not clear. By mass and velocity definite

quantities may be understood ; not so, however, by

THE EXTENSION OF THE PRINCIPLES. 365

space, when the time is not stated in which the space
is described. If, however, unit of time be meant, the
distinction of space and velocity in the examples treated
by Maupertuis is, to say the least, peculiar. It appears
that Maupertuis reached this obscure expression by an
unclear mingling of his ideas of vis viva and the prin-
ciple of virtual velocities. Its indistinctness will be
more saliently displayed by the details.

2. Let us see how Maupertuis applies his principle. Determina-

If My m be two inelastic masses, Cand c their velocities laws of im-
pact by this

before impact, and u their common velocity after im- principle,
pact, Maupertuis requires, (putting here velocities for
spaces,) that the "action" expended in the change of
the velocities in impact shall be a minimum. Hence,
M{C — u)^ '\- m{c — «) 2 is a minimum \ that is,
m\c— ») + w (r — «) == 0 ; or

MC -\- mc

For the impact of elastic masses, retaining the same
designations, only substituting Kand v for the two ve-
locities after impact, the expression M{C — ^)^ +
in{c — ^^7^)2 is a minimum; that is to say,

M{^C—V)dV^m{c — v^dv = ^ (1)

In consideration of the fact that the velocity of ap-
proach before impact is equal to the velocity of reces-
sion after impact, we have

C—c = — (^V—v) or

C-^ r-(r + rO = 0 (2)

and

d V— dv = 0 (3)

The combination of equations (i), (2), and (3)
readily gives the familiar expressions for ^ and v.
These two cases may, as we see, be viewed as pro-

366 THE SCIENCE OF MECHANICS,

cesses in which the least change of vis viva by reaction
takes place, that is, in which the Uast counter-work is
done. They fall, therefore, under the principle of
Gauss.
Maupcr- 3. Peculiar is Maupertuis's deduction of the law of

tuis;sde- •' . _.

ductionof the lever. Two masses M and m (Fig. 188) rest on a

the law of . . .

the lever by bar <z, which the fulcrum divides into the portions

this pnn- 1 1 •

cipie. X and a — jc. If the bar be set in rotation^ the veloci-
ties and the spaces described will be proportional to
the lengths of the lever-arms, and Mx^ •\- m{a — x)'^
is the quantity to be made a minimum, that is Mx —
m{a — jc) = 0 ; whence x = majM + /«, — a condition

that in the case of equilib-

, rium is actually fulfilled. In

^' • -w criticism of this, it is to be

^ ^_^ remarked, first, that masses

Fig. 188. not subject to gravity or

other forces, as Maupertuis
here tacitly assumes, are always in equilibrium, and,
secondly, that the inference from Maupertuis*s deduc-
tion is that the principle of least action is fulfilled
only in the case of equilibrium, a conclusion which it
was certainly not the author's intention to demonstrate.
The correc- If it Were sought to bring this treatment into ap-
pertuis's proximatc accord with the preceding, we should have
to assume that the heavy masses M and m constantly
produced in each other during the process the least
possible change of vis viva. On that supposition, we
should get, designating the arms of the lever briefly by
tf, b^ the velocities acquired in unit of time by u^ v, and
the acceleration of gravity by g, as our minimum ex-
pression, M{g — «)2 -|- m{g — v)^\ whence Af{g — «)
du + m(^g — v)dv = 0. But in view of the connection
of the masses as lever,

deduction.

THE EXTENSION OF THE PRINCIPLES.

367

u
a

whence these equations correctly follow

Afa — md Afa — mfi

and for the case of equilibrium, where u = v = 0,

Ma — mb = 0.

Thus, this deduction also, when we come to rectify
it, leads to Gauss's principle.

4. Following the precedent of Fermat and Leib- Trcatmeni

of the mo-

nitz, Maupertuis also treats by his method the motion rAonoiw^x

by the pnn-

of light. Here again, however, cfpie of

• he employs the notion *' least ac-
tion " in a totally different sense.
The expression which for the
case of refraction shall be a min-
imum, is m , AR + « • RB^
where AR and RB denote the
paths described by the light in
the first and second media re-
spectively, and m and n the corresponding velo-
cities. True, we really do obtain here, if R be de-
termined in conformity with the minimum condition,
the result sma /sin ft = n/m =^ const. But before, the
* ' action " consisted in the change of the expressions
mass X velocity X distance ; now, however, it is con-
stituted of the sum of these expressions. Before, the
spaces described in unit of time were considered ; in
the present case the total spaces traversed are taken.
Should not m, AR — n. RB or [m — n){AR — RB)
be taken as a minimum, and if not, why not? But

Fig. 189.

368 THE SCIENCE OF MECHANICS.

even if we accept Maupertuis's conception, the recip-
rocal values of the velocities of the light are obtained,
and not the actual values,
characteri- It wiU thus be Seen that Maupertuis really had no
Manper- principle, properly speaking, but only a vague form-
cipie. ula, which was forced to do duty as the expression of
different familiar phenomena not really brought under
one conception. I have found it necessary to enter
into some detail in this matter, since Maupertuis's per-
formance, though it has been unfavorably criticised by
all mathematicians, is, nevertheless, still invested with
a sort of historical halo. It would seem almost as if
something of the pious faith of the church had crept
into mechanics. However, the mere endeavor to gain
a more extensive view, although beyond the powers of
the author, was not altogether without results. Euler,
at least, if not also Gauss, was stimulated by the at-
tempt of Maupertuis.
Euier'scon- 5. Euler's vicw is, that the purposes of the phe-

tributions tt ■% j • r

to this sub- nomena of nature afford as good a basis of explana-

ject. ,

tion as their causes. If this position be taken, it will
be presumed a priori that all natural phenomena pre-
sent a maximum or minimum. Of what character this
maximum or minimum is, can hardly be ascertained
by metaphysical speculations. But in the solution of
mechanical problems by the ordinary methods, it is
possible, if the requisite attention be bestowed on the
matter, to find the expression which in all cases is
made a maximum or a minimum. Euler is thus not
led astray by any metaphysical propensities, and pro-
ceeds much more scientifically than Maupertuis. He
seeks an expression whose variation put = 0 gives the
ordinary equations of mechanics.

For a single body moving under the action of forces

THE EXTENSION OF THE PRINCIPLES. 369

Euler finds the requisite expression in the formula The form

which the

Cv dsy where ds denotes the element of the path and principle
V the corresponding velocity. This expression is sm aller Euier's
for the path actually taken than for any other infinitely
adjacent neighboring path between the same initial
and terminal points, which the body may be constrained
to take. Conversely, therefore, by seeking the path that
makes Cv ds a minimum, we can also determine the
path. The problem of minimising Cv d's is, of course,
as Euler assumed, a permissible one, only when v de-
pends on the position of the elements ds, that is to
say, when the principle of vis viva holds for the forces,
or a force-function exists, or what is the same thing,
when Z7 is a simple function of coordinates. For a mo-
tion in a plane the expression would accordingly as-
sume the form

^ . , c^y

In the simplest cases Euler 's principle is easily veri-
fied. If no forces act, v is constant, and the curve of
motion becomes a straight line, for which Cvds^=^
V C ds is unquestionably shorter than for any other
curve between the same terminal points.
Also, a body moving on a curved surface
without the action of forces or friction,
preserves its velocity, and describes on
the surface a shortest line.

The consideration of the motion of a | \^ euIct's

projectile in a parabola ^^C (Fig. 190) n^ SppulS^to

will also show that the quantity Jv ds d AC S^Trofec-

is smaller for the parabola than for any „. ****'

Other neighboring curve ; smaller, even,
than for the straight line ABC between the same ter-
minal points. The velocity, here, depends solely on the

370 THE SCIENCE OF MECHANICS,

Mathemat- vertical space described by the body, and is therefore

ical dcvel- j -j *

opmentof the Same for all curves whose altitude above OC is the
same. If we divide the curves by a system of horizontal
straight lines into elements which severally correspond,
the elements to be multiplied* by the same t^'s, though
in the upper portions smaller for the straight line AD
than for AB, are in the lower portions just the reverse ;
and as it is here that the larger v^s come into play, the
sum upon the whole is smaller iox ABC than for the
straight line.

Putting the origin of the codrdinates at Ay reckon-
ing the abscissas x vertically downwards as positive,
and calling the ordinates perpendicular thereto y^ we
obtain for the expression to be minimised

where g denotes the acceleration of gravity and a the
distance of descent corresponding to the initial velocity.
As the condition of minimum the calculus of variations
gives

= C or

' + m

dx ^ —^ -' °'

Vlg\a + *) - C»

/Cdx
l/2g(a + x)-^^C^
and, ultimately,

y =

y=-\/2g(a-^x)—C^^C\

THE EXTENSION OF THE PRINCIPLES, 371

where C and C* denote constants of integration that
pass into C=^V 7,ga and C= 0, if for jt = 0, dx/dy = 0
and j^ = 0 be taken. Therefore, y = 21/ ax. By this
method, accordingly, the path of a projectile is shown
to be of parabolic form.

6.- Subsequently, Lagrange drew express attention The addi-
to the fact that Euler's principle is applicable only in Rrange and
cases in which the principle of vis viva holds. Jacobi
pointed out that we cannot assert that Cv ds for the ac-
tual motion is a minimum^ but simply that the variation of
this expression, in its passage fo an infinitely adjacent
neighboring path, is = 0. Generally, indeed, this con-
dition coincides with a maximum or minimum, but it
is possible that it should occur without such ; and the
minimum property in particular is subject to certain
limitations. For example, if a body, constrained to
move on a spherical surface, is set in motion by some
impulse, it will describe a great circle, generally a
shortest line. But if the length of the arc described
exceeds 1 80°, it is easily demonstrated that there exist
shorter infinitely adjacent neighboring paths between
the terminal points.

7. So far, then, this fact only has been pointed out, Baler's,
that the ordinary equations of motion are obtained by but one of
equating the variation of Cv ds to zero. But since the give^he *
properties of the motion of bodies or of their paths may oJmoUon.
always be defined by differential expressions equated
to zero, and since furthermore the condition that the
variation of an integral expression shall be equal to
zero is likewise given by differential expressions equated
to zero, unquestionably various other integral expres-
sions may be devised that give by variation the ordi-
nary equations of motion, without its following that the

372 THE SCIENCE OF MECHANICS,

integral expressions in question must possess on that
account any particular physical significance.
Yet the ex- 8. The Striking fact remains, however, that so simple
must pos- an expression as fr ds does possess the property men-
icai import, tioued, and we will now endeavor to ascertain its phys-
ical import. To this end the analogies that exist be-
tween the motion of masses and the motion of light, as
well as between the motion of masses and the equilib-
rium of strings — analogies noted by John Bernoulli
and by Mobius — will stand us in stead.

A body on which 'ho forces act, and which there-
fore preserves its velocity and direction constant, de-
scribes a straight line. A ray of light passing through
a homogeneous medium (one having everywhere the
same index of refraction) describes a straight line. A
string, acted on by forces at its extremities only, as-
sumes the shape of a straight line.
Elucidation A body that moves in a curved path from a point

Of this im- • r» « 1 1 •

port by the ^ to a point B and whose velocity 7f =^ fl>(^, y, z) is a

motion of a . , ,

mass, the function of Coordinates, describes between A and B a

motion of a , • i n /• , • • • a

ray of light, curve f or which generally H' as is a minimum. A ray
eauiiibrium of light passing from A to B describes the same curve,
if the refractive index of its medium, « = ^(jc, y, z),
is the same function of coordinates ; and in this case
fnds is a minimum. Finally, a string passing from
A to B will assume this curve, if its tension S =
<P (p^f y* ^) is the same above-mentioned function of co-
ordinates ; and for this case, also, CSds is a minimum.
The motion of a mass may be readily deduced from
the equilibrium of a stringy as follows. On an element
ds oidL. string, at its two extremities, the tensions S, S'
act, and supposing the force on unit of length to be P^
in addition a force P. ds. These three forces, which
we shall represent in magnitude and direction by BA,

THE EXTENSION OF THE PRINCIPLES.

373

BC^ BD (Fig. 191), are in equilibrium. If now, a body, The moUon
with a velocity ly represented in magnitude and diiec- deduced
tion by AB^ enter the element of the path ds^ and re- eauiiibriun
ceive within the same the velocity component BF =
— BD, the body will proceed on-
ward with the velocity v' = BC.
Let Q be an accelerating force
whose action is directly opposite
to that of B; then for unit of time
the acceleration of this force ^11
be Q, for unit of length of the
string Q/v, and for the element
of the string (Q/v)ds, The body will move, therefore,
in the curzfe of the string, if we establish between the
forces P and the tensions S, in the case of the string,
and the accelerating forces Q and the velocity v in the
case of the mass, the relation

Fig. Z9Z.

The minus sign indicates that the directions of P and
Q are opposite.

A closed circular string is in equilibrium when be- The eqm-

, librinm ox

tween the tension S of the strmg, everywhere constant, closed

strings.

and the force P falling radially outwards on unit of
length, the relation P = Sjr obtains, where r is the
radius of the circle. A body will move with the con-
stant velocity v in a circle, when between the velocity
and the accelerating force Q acting radially inwards
the relation

= - or ^ = - obtains.
V r r

A body will move with constant velocity v in any curve
when an accelerating force Qz=v^/r constantly acts

374

THE SCIENCE OF MECHANICS.

on it in the direction of the centre of curvature of each
element. A string will lie under a constant tension S
in any curve if a force P = Sjr acting outwardly from
the centre of curvature of the element is impressed on
unit of length of the string.
The deduc* No concept aualogous to that of force is applicable
motion of to the tnotioTi of light. Consequently, the deduction of

light froni

the motions the motion of light from the equilibrium of a string or

of masses ,

and the, the motion of a mass must be differently effected. A

equilibnnm , , ,

of strings, mass, let us say, is moving with the velocity AB = v.

(Fig. 192.) A force in the direction
B£> is impressed on the mass which
produces an increase of velocity B£,
so that by the composition of the ve-
locities BC = AB and B£ the new
velocity BP = v' is produced. If we
resolve the velocities v, v* into com-
ponents parallel and perpendicular to
the force in question, we shall per-
ceive that the parallel components alone
are changed by the action of the force.
This being the case, we get, denoting
by k the perpendicular component, and by a and c^
the angles v and v* make with the direction of the

Fig. 193.

force.

k^=v sin a
k = 7/ sin a' or

sm^
sin a*

V
V

If, now, we picture to ourselves a ray of light that
penetrates in the direction of v a refracting plane at
right angles to the direction of action of the force, and
thus passes from a medium having the index of refrac-

THE EXTENSION OF THE PRINCIPLES, 375
tion n into a medium having the index of refraction ri^ Develop-

, mentor this

where «/«' = 7^7^', this ray of light will describe the illustration,
same path as the body in the case above. If, there-
fore, we wish to imitate the motion of a mass by the
motion of a ray of light (in the same curve), we must
everywhere put the indices of refraction, «, proportional
to the velocities. To deduce the indices of refraction
from the forces, we obtain for the velocity

// I -^ 1 = Pdq, and
for the index of refraction, by analogy.

where P denotes the force and dg a distance-element
in the direction of the force. If ds is the element of
the path and a the angle made by it with the direction
of the force, we have then

d[ (,-] = Pcosa. ds

(n^\
d\ ] = Pcosa. ds.

For the path of a projectile, under the conditions above
assumed, we obtained the expression y = 2yax. This
same parabolic path will be described by a ray of light,
if the law n == \/2g(a -}- x) be taken as the index of
refraction of the medium in which it travels.

9. We will now more accurately investigate the Relation of
manner in which this minimum property is related to mum prop-
the/^r»f of the curve. Let us take, first, (Fig. 193) afoimof
broken straight line ABC, which intersects the straight
line MJV, put AB = s, BC = s', and seek the condition
that makes vs -f 7'V a minimum for the line that passes

376

7'HE SCIENCE OF MECHANICS,

First, de-
duction of
the mini-
mum condi'
tion.

through the fixed points A and B^ where v and v' are
supposed to have different, though constant, values
above and below MN, If we displace the point B an
infinitely small distance to D^ the new line through A
and C will remain parallel to the original one, as the
drawing symbolically shows. The expression vs -\- v's'
is increased hereby by an amount

— timsma + v'ms\na\

where m ^= DBj or by an amount — v sin a + 7/ sin or'.
The condition of the minimum, consequently, is that

— V sin a + v' sin or' = 0

s\na

or -T ,=

smar

7f
V

R N

FiR. 193.

ri«. 19,.

If the expression sjv -{■ s ji^' is to be made a minimum,
we have, in a similar way,

/•

smo: V

sin «' V*

Second, the If, next, we Consider the case of a string stretched

SfoiU^con*? in the direction ABC, the tensions of which .Sand ^S"

eauiiibrium are different above and below MN, in this case it is

the minimum of Ss -f S*s* that is to be dealt with. To

obtain a distinct idea of this case, we may imagine the

THE EXTENSION OF THE PRINCIPLES.

motion of a
ray of light.

String stretched once between A and B and thrice be-
tween B and C, and finally a weight P attached. Then
S= P and 5' = 3 -P. If we displace the point B a dis-
tance m, any diminution of the expression Ss -{- S's'
thus effected, will express the increase of work which
the attached weight P performs. If — Sm sin a -^
S'm sin a' = 0, no work is performed. Hence, the mini-
mum of Ss -\- S's' corresponds to a maximum of work.
In the present case the principle of least action is sim-
ply a different form of the principle of virtual displace-
ments.

Now suppose that ABC is a ray of light, whose ve- Third, the

application

locities V and v' above and below MN are to each other of thi« con-
dition to the
as 3 to I. The motion of light be-
tween two points A and B is such
that the light reaches ^ in a mini-
mum of time. The physical reason
of this is simple. The light travels
from A to B^ in the form of ele-
mentary waves, by different routes.
Owing to the periodicity of the light,
the waves generally destroy each
other, and only those that reach the
designated point in equal times, that is, in equal phases,
produce a result. But this is true only of the waves
that arrive by the minimum path and its adjacent neigh-
boring paths. Hence, for the path actually taken by
the light s/v + j'/p' is a minimum. And since the in-
dices of refraction n are inversely proportional to the
velocities v of the light, therefore also ns ■\- n*s* is a
minimum.

In the consideration of the motion of a mass the con-
dition that vs -\- v's* shall be a minimum, strikes us as
something novel. (Fig. 195.) If a mass, in its passage

Fig- 195-

378 THE SCIENCE OF MECHANICS,

Fourth, it:! through a plane MN^ receive, as the result of the action
fo the m<^" of a force impressed in the direction DB, an increase of
mass. velocity, by which v, its original velocity, is made v' y we
have for the path actually taken by the mass the equa-
tion 7/ sin a = z/' sin a' = k. This equation^ which is also
the condition of minimum, simply states that only the ve-
locity-component parallel to the direction of the force is
altered, but that the component k at right angles thereto re-
mains unchanged. Thus, in this case also, Euler's prin-
ciple simply states a familiar fact in a new form.
Formofthe lo. The minimum condition — t^sinar+ e^'sina'sTrO
condition may also be written, if we pass from a finite broken
to curves. Straight line to the elements of curves, in the form

— V sin a -\- (y -\- dv) sin(a + dd) = 0
or

d(v sin «) = 0

or, finally,

tf sin a = const.

In agreement with this, we obtain for the motion

of light

d {n sin «) = 0, n sin a = const,

\ ^' J V

and for the equilibrium of a string

d{Ssmd) = 0, ^sina = const.

To illustrate the preceding remarks by an ex-
ample, let us take the parabolic path of a projectile,
where a always denotes the angle that the element of
the path makes with the perpendicular. Let the ve-
locity he 7f = \/2g(^a -\- x), and let the axis of the_y-or-
dinates be horizontal. The condition v. sin a = const,
or V 2g{a -f- A*) . dy/ds = const, is identical with that
which the calculus of variation gives, and we now know

THE EXTENSION OF THE PRINCIPLES,

yi^

Fig. xgC

its xww^/^/^w/Va/ significance. If we picture to ourselves niustration

. , T . / >- of the three

a string whose tension is 5 = k 2r (« + x\ an arrange- typical

... caaea by

ment which might be effected by fixing frictionless curvilinear

. . motiona.

pulleys on horizontal parallel rods placed in a vertical
plane, then passing the string through these a sufficient
number of times, and finally attaching
a weight to the extremity of the string,
we shall obtain again, for equilibrium,
the preceding condition, the phys-
ical significance of which is now ob-
vious. When the distances between
the rods are made infinitely small the
string assumes the parabolic form.
In a medium, the refractive index of
which varies in the vertical direction
by the law n = \^2g{a + x), or the velocity of light in
which similarly varies by the law v = \/\/'2g{a + x)^
a ray of light will describe a path which is a parabola.
If we should make the velocity in such a medium
V = \/2g{a-\-x), the ray would describe a cycloidal path,
for which, not CV2g{a + x). ds, but the expression
Cds/\/2g{a + x) would be a minimum.

II. In comparing the equilibrium of a string with
the motion of a mass, we may employ in place of a
string wound round pulleys,
a simple homogeneous cord,
provided we subject the cord
to an appropriate system of
forces. We readily observe
that the systems of forces
that make the tension, or,
as the case may be, the ve-
locity, the same function of coordinates, are differ-
ent. If we consider, for example, the force of gravity.

Fig. 197.

38o THE SCIENCE OF MECHANICS.

The condi- V = V igia + x\ A String, howcver, subjected to the

tionsand ... - -x • t

conse- action of gravity, forms a catenary, the tension ot
Oiepreced- which IS given by the formula S = m — nx^ where m

ing analo-
gies- and n are constants. The analogy subsisting between

the equilibrium of a string and the motion of a mass is
substantially conditioned by the fact that for a string
subjected to the action of forces possessing a force-
function U, there obtains in the case of equilibrium
the easily demonstrable equation U -\- S= const. This
physical interpretation of the principle of least action
is here illustrated only for simple cases ; but it may
also be applied to cases of greater complexity, by
imagining groups of surfaces of equal tension, of equal
velocity, or equally refractive indices constructed which
divide the string, the path of the motion, or the path
of the light into elements, and by making a in such a
case represent the angle which these elements make
with the respective surface- normals. The principle of
least action was extended to systems of masses by La-
grange, who presented it in the form

62m Czfds = 0.

If we reflect that the principle of vis viva, which is the
real foundation of the principle of least action, is not
annulled by the connection of the masses, we shall
comprehend that the latter principle is in this case also
valid and physically intelligible.

IX.

Hamilton's principle.

I. It was above remarked that various expressions
can be devised whose variations equated to zero give
the ordinary equations of motion. An expression of
this kind is contained in Hamilton's principle

THE EXTENSION OF THE PRINCIPLES. 381

S f(C/'+ T) dt = 0, or The points

J ^ ' ^ ' of identity

'0 of Hamil-

tx ton's and

/q ciples.

where <^C/'and c^T' denote the variations of the work
and the vis viva^ vanishing for the initial and terminal
epochs. Hamilton's principle is easily deduced from
D'Alembert's, and, conversely, D'Alembert's from
Hamilton's ; the two are in fact identical, their differ-
ence being merely that of form. *

2. We shall not enter here into any extended in- Hamilton's

. . pnncipie

vestigation of this subject, but simply exhibit the iden- applied to

° . . •* ' '^ •* the motion

tity of the two principles by an example — of a wheel

the same that served to illustrate the prin-
ciple of D'Alembert : the motion of a wheel
and axle by the over-weight of one of its
parts. In place of the actual motion, we
may imagine, performed in the same inter-
val of time, a different motion, varying in-
finitely little from the actual motion, but p. ^^
coinciding exactly with it at the beginning
and end. There are thus produced in every element
of time dtf variations of the work (SU) and of the vis
viva (^ST); variations, that is, of the values C/'and T
realised in the actual motion. But for the actual mo-
tion, the integral expression, above stated, is = 0, and
may be employed, therefore, to determine the actual
motion. If the angle of rotation performed varies in
the element of time di an amount a from the angle of
the actual motion, the variation of the work corre-
sponding to such an alteration will be

6C/= {PR —Qr)a = Ma.

* Compare, for example, Kirchhoff, VarUtungtn Mber matkematitche Pky-
siky Meekanik, p. aj et teqg.^ and Jacobi, VorUsungtn Mhtr DynamiA, p. 58.

382 THE SCIENCE OF MECHANICS.

Mathemat- The vis viva, for any given angular velocity a?, is

ical devel-
opment of ,^ 1 X ^ ^^ ^ 00)^
tfiiscase. T=^ - {PR'^ + ^/-a) --,

and for a variation dco of this velocity the variation of
the vis viva is

But if the angle of rotation varies in the element <// an
amount a,

dcD:^-- and

The form of the integral expression, accordingly, is

fUa + N'^''

<// = 0.

But as

d ,j.^ . dN da

therefore,

The second term of the left-hand member, though,
drops out, because, by hypothesis, at the beginning
and end of the motion a = 0. Accordingly, we have

an expression which, since a in every element of time
is arbitrary, cannot subsist unless generally

-- '1= »■

THE EXTENSION OF THE PRINCIPLES. 383

Substituting for the symbols the values they represent,
we obtain the familiar equation

//«_ PR—Qr
d't '^ PR^ ^r Qr^^'
D'Alembert*s principle gives the equation The same

results ob-
'^ (iN\ . Uinedby

/ bert's prin-

which holds for every possible displacement. We might, *^*^*®'
in the converse order,. have started from this equation,
have thence passed to the expression

/(^_^)«^, = 0.

and, finally, from the latter proceeded to the same re-
sult

f(Ma + JSr^] di = 0.

3. As a second and more simple example let us ninstration
consider the motion of vertical descent. .For every Sy the rnc^
infinitely small displacement s the equation subsists ticai de-

SCCDt

[ntg — m(^dv/diy]s = 0f in which the letters retain
their conventional significance. Consequently, this
equation obtains

w VT ]s .di = 0,

which, as the result of the relations

'1
f[mg -

.(mvs) dv , ds ^

384 THE SCIENCE OF MECHANICS.

J. fit = (m 7' s) = 0,

provided ^ vanishes at both limits, passes into the form

mgs + mv ,\tf/= 0,

to ^
that is, into the form of Hamilton's principle.

Thus, through all the apparent differences of the
mechanical principles a common fundamental same-
ness is seen. These principles are not the expression
of different facts, but, in a measure, are simply views
of different aspects of the same fact.

SOME APPLICATIONS OF THE PRINCIPLES OF MECHANICS TO
HYDROSTATIC AND HYDRODYNAMIC QUESTIONS.

Method of I. We will now supplement the examples which

the action we have given of the application of the principles

of Ersivitv

onliquid of mechanics, as they applied to rigid bodies, by a

xnflsscs.

few hydrostatic and hydrodynamic illustrations. We
shall first discuss the laws of equilibrium of a weightless
liquid subjected exclusively to the action of so-called
molecular forces. The forces of gravity we neglect in
our considerations. A liquid may, in fact, be placed
in circumstances in which it will behave as if no forces
of gravity acted. The method of this is due to Pla-
teau.* It is effected by immersing olive oil in a mix-
ture of water and alcohol of the same density as the
oil. By the principle of Archimedes the gravity of the
masses of oil in such a mixture is exactly counterbal-
anced, and the liquid really acts as if it were devoid of
weight.

* Statigue expirimentale et thiorique des Itqutdet, 1873.

THE EXTENSION OF THE PRINCIPLES. 385

2. First, let us imagine a weightless liquid mass The work of
free in space. Its molecular forces, we know, act only forces de-

.... rry \ * i» !*• pcndcnt on

at very small distances. Takmg as our radius the dis- a change in

, . , , , , r theliauid's

tance at which the molecular forces cease to exert a superficial

area.

measurable influence, let us describe about a particle
fl, ^, c in the interior of the mass a sphere — the so-
called sphere of action. This sphere of action is regu-
larly and uniformly filled with other particles. The
resultant force on the central particles a^ b^ c is there-
fore zero. Those parts only that lie at a distance from
the bounding surface less than the radius of the sphere
of action are in different dynamic conditions from the
particles in the interior. If the radii of curvature of

Fig. 199- F»K- aoo.

the surface-elements of the liquid mass be all regarded
as very great compared with the radius of the sphere
of action, we may cut off from the mass a superficial
stratum of the thickness of the radius of the sphere of
action in which the particles are in different physical
conditions from those in the interior. If we convey
a particle a in the interior of the liquid from the posi-.
tion a to the position b or c^ the physical condition
of this particle, as well as that of the particles which
take its place, will remain unchanged. No work can
be done in this way. Work can be done only when a
particle is conveyed from the superficial stratum into
the interior, or, from the interior into the superficial
stratum. That is to say, work can be done only by a

386

THE SCIENCE OF MECHANICS.

change of size of the surface. The consideration whether
the density of the superficial stratum is the same as
that of the interior, or whether it is constant through-
out the entire thickness of the stratum, is not primarily
essential. As will readily be seen, the variation of the
surface-area is equally the condition of the perform-
ance of work when the liquid mass is immersed in a
second liquid, as in Plateau's experiments.
Diminurion We now inquire whether the work which by the

of super- , - ... ... „

ficiai area^ transportation of particles mto the interior effects a
tivework. diminution of the surface-area is positive or negative,
that is, whether work is performed or work is ex-
pended. If we put two fluid drops in contact, they

will coalesce of their own accord;
and as by this action the area
of the surface is diminished, it
follows that the work that pro-
duces a diminution of superfi-
Fig. aoi. ^^^^ ^TQ^. in a liquid mass is posi-

tive. Van der Mensbrugghe has
demonstrated this by a very pretty experiment. A
square wire frame is dipped into a solution of soap and
water, and on the soap-film formed a loop of moistened
thread is placed. If the film within the loop be punc-
tured, the film outside the loop will contract till the
thread bounds a circle in the middle of the liquid sur-
face. But the circle, of all plane figures of the same
circumference, has the greatest area ; consequently,
the liquid film has contracted to a minimum,
conscguent The following wiU now be clear. A weightless

condition , . , .

of liquid Hquid, the forces acting on which are molecular forces,

equiubrinm , , . . .

will be in equilibrium in all forms in which a system of
virtual displacements produces no alteration of the
liquid's superficial area. But all infinitely small changes

THE EXTENSION OF THE PRINCIPLES 387

of form may be regarded as virtual which the liquid
admits without alteration of its volume. Consequently,
equilibrium subsists for all liquid forms' for which an
infinitely small deformation produces a superficial va-
riation = 0. For a given volume a minimum of super-
ficial area gives stable equilibrium; a maximum un-
stable equilibrium.

Among all solids of the same volume, the sphere
has the least superficial area. Hence, the form which
a free liquid mass will assume, the form of stable equi-
librium, is the sphere. For this form a maximum of
work is done ; for it, no more can be done If the
liquid adheres to rigid bodies, the form assumed is de-
pendent on various collateral conditions, which render
the problem more complicated.

3. The connection between the sise and the/i^rm of Hodeofde-
the liquid surface may be investigated as follows. We the connec-
tmagine the closed outer sur- liieand

face of the liquid to receive « liquid lu-

without alteration of the li-
quid's volume an infinitely
small variation. By two sets of
mutually perpendicular lines
of curvature, we cut up the

original surface into infinitely small rectangular ele-
ments. At the angles of these elements, on the original
surface, we erect normals to the surface, and determine
thus the angles of the corresponding elements of the
varied surface. To every element dO of the original
surface there now corresponds an element dO' of the
varied surface ; by an infinitely small displacement, Sn,
along the normal, outwards or inwards, t/0 passes into
40' and into a corresponding variation of magnitude.

Let dfi, dq be the sides of the element dO. For the

388

THE SCIENCE OF MECHANICS,

The mathe- sides dp\ dq' of the element dO\ then, these relations

velopment obtain
of this

method. f ^ ^ Sn

dp' = dp\\+-^-

dq' = dg[l-{-

where r and r are the radii of curvature of the princi-
pal sections touching the elements of the lines of cur-
vature /, q, or the so-called principal radii of curva-
ture. * The radius of curvature of an outwardly convex
element is reckoned as positive, that of an outwardly
concave element as negative, in the usual manner. For
the variation of the element we obtain, accordingly.

6.dO = dO' — dO = dpdg[l+~

6 n

Neglecting the higher powers of <y« we
get

6.dO = y +~A6n,dO,

The variation of the whole surface,
then, is expressed by

=/(i+i

Sn.dO

(1)

Fig. 203.

Furthermore, the normal displacements
must be so chosen that

J6n.dO = ^ (2)

that is, they must be such that the sum of the spaces
produced by the outward and inward displacements of

* The normal at any point of a surface is cut by normals at infinitely neigh-
boring points that lie in two directions on the surface from the original point,
these two directions being at right angles to each other ; and the distances
from the surface at which these normals cut are the two principal, or extreme,
radii of curvature of the surface. — Trans.

THE EXTENSION OF THE PRINCIPLES, 389

the superficial elements (in the latter case reckoned as
negative) shall be equal to zero, or the volume remain
constant.

Accordingly, expressions (i) and (2) can be put a condition

on whicli

simultaneously = 0 only if i/r + i/r' \i^s the same value ^^ffiXiet7L\-
for all points of the surface. This will be readily seen pressions
from the following consideration. Let the elements depends.'
dO of the original surface be symbolically represented
by the elements of the line AX (Fig. 204) and let the
normal displacements (^/r be erected as ordinates
thereon in the plane E^ the outward displacements up-
wards as positive and the inward displacements down-
wards as negative.
Join the extremities E

of these ordinates so
as to form a curve,
and take the quadra-
ture of the curve,
reckoning the sur-
face above AX sls positive and that below it as nega-
tive. For all systems oi dn for which this quadra-
ture = 0, the expression (2) also := 0, and all such
systems of displacements are admissible, that is, are
virtual displacements.

Now let us erect as ordinates, in the plane E', the
values of i /^ + i /^ that belong to the elements dO. A
case may be easily imagined in which the expressions
(i) and^2) assume coincidently the value zero. Should,
however, i /r -f ^/^ have different values for different
elements, it will always be possible without altering
the zero-value of the expression (2), so to distribute
the displacements 6n that the expression (i) shall be
different from zero. Only on the condition that i/r -\-
i/r' has the same value for all the elements, is expres-

Fig. ao4.

390 THE SCIENCE OF MECHANICS,

sion (i) necessarily and universally equated to zero
with expression (2).
The sum Accordingly, from the two conditions (i) and (2) it

which for , ,, i ^ / , ^ / , i • ,

equilibrium follows that l/r + 1/r = const \ that IS to say, the sum
constant for of the reciprocal values of the principal radii of curva-
surface. ture, or of the radii of curvature of the principal nor-
mal sections, is, in the case of equilibrium, constant
for the whole surface. By this theorem the dependence
of the area of a liquid surface on its superficial /^rm is
defined. The train of reasoning here pursued was
first developed by Gauss,* in a much fuller and more
special form. It is not difficult, however, to present
its essential points in the foregoing simple manner.
Application 4. A liquid mass, left wholly to itself, assumes, as

of this gen-

erai condi- we have Seen, the spherical form, and presents an ab-
interrupted solute minimum of superficial area. The equation

liquid mas- , , . ,

ses. V'' + 1/r' = const is here visibly fulfilled in the form

7.IR = const, Jl being the radius of the sphere. If the
free surface of the liquid mass be bounded by two solid
circular rings, the planes of which are parallel to each
other and perpendicular to the line joining their mid-
dle points, the surface of the liquid mass will assume
the form of a surface of revolution. The nature of the
meridian curve and the volume of the enclosed mass
are determined by the radius of the rings J?, by the
distance between the circular planes, and by the value
of the expression 1/r + V''' ^^^ *^® surface of revolu-
tion. When

r^ r' r ^00 ""7?'

the surface of revolution becomes a cylindrical surface.
For 1/r + l/r'= 0, where one normal section is con-

* Principia Generalia Theoria Figura Flnidorutn in Statu /Sguilibrii,
GOttingen, 1830 ; Werkt, Vol. V, 29, GAttingen, 2867.

THE EXTENSION OF THE PRINCIPLES. 391

vex and the other concave, the meridian curve assumes
the form of the catenary; Plateau visibly demonstrated
these cases by pouring oil on two circular rings of wire
fixed in the mixture of alcohol and water above men-
tioned.

Now let us picture to ourselves a liquid mass Liquid mas-
ses whose
bounded by surface-parts for which the expression surfaces are

partly con-

1/r -f lA' has a positive value, and by other parts cave and
for which the same expression has a negative value, vex.
or, more briefly expressed, by convex and concave sur-
faces. It will be readily seen that any displacement
of the superficial elements outwards along the normal
will produce in the concave parts a diminution of the
superficial area and in the convex parts an increase.
Consequently, work is performed when concave surfaces
move outwards and convex surfaces inwards. Work
also is performed when a superficial portion moves
outwards for which 1/r -|- 1/r' = -|- a, while simulta-
neously an equal superficial portion for which 1 /r -f
1/r' > tf moves inwards.

Hence, when differently curved surfaces bound a
liquid mass, the convex parts are forced inwards and
the concave outwards till the condition 1/r + l/r' =
const is fulfilled for the entire surface. Similarly, when
a connected liquid mass has several isolated surface-
parts, bounded by rigid bodies, the value of the ex-
pression 1/r + 1/r' must, for the state of equilibrium
be the same for all free portions of the surface.

For example, if the space between the two circular Experi-
rings in the mixture of alcohol and water above re- illustration
ferred to, be filled with oil, it is possible, by the use conditions.
of a sufficient quantity of oil, to obtain a cylindrical
surface whose two bases are spherical segments. The
curvatures of the lateral and basal surfaces will accord-

392 THE SCIENCE OF MECHANICS.

ingly fulfil the condition 1/^ -|- l/oo = 1/p -f l/p> or
p = 2R^ where p is the radius of the sphere and R that
of the circular rings. Plateau verified this conclusion
by experiment.
Liquidmas- 5. Let US now studv a weightless liquid mass which
ing a hoi- eucloses a hollow space. The condition that 1/r -(- 1/r'

low space. , ,, , , 1 r 1 • • •%

shall have the same value for the mtenor and exterior
surfaces, is here not realisable. On the contrary, as
this sum has always a greater positive value for the
closed exterior surface than for the closed interior sur-
face, the liquid will perform work, and, flowing from
the outer to the inner surface, cause the hollow space
to disappear. If, however, the hollow space be occu-
pied by a fluid or gaseous substance subjected to a de-
terminate pressure, the work done in the last-men-
tioned process can be counteracted by the work ex-
pended to produce the compression, and thus equilib-
rium may be produced.
The me- Let US picture to ourselves a liquid mass confined

chaDical

properties between two similar and similarly situated surfaces

very near each other. A bubble is such
a system. Its primary condition of equi-
librium is the exertion of an excess of
pressure by the inclosed gaseous con-
tents. If the sum 1/r -|- 1/r' has the
value -4- a for the exterior surface, it will
have for the interior surface very nearly
the value — a, A bubble, left wholly to itself, will al-
ways assume the spherical form. If we conceive such
a spherical bubble, the thickness of which we neglect,
the total diminution of its superficial area, on the
shortening of the radius r by dr^ will be 16 rndr. If,
therefore, in the diminution of the surface by unit
of area the work A is performed, then A .i6mdr will

THE EXTENSION OF THE PRINCIPLES, 393

be the total amount of work to be compensated for
by the work of compression /.4r2;r//r expended by
the pressure / on the inclosed contents. From this
follows ^Ajr :=/ ; from which A may be easily calcu-
lated if the measure of r is obtained and / is found by
means of a manometer introduced in the bubble.

An open spherical bubble cannot subsist. If an open

bubbles.

open bubble is to become a figure of equilibrium, the
sum 1/r 4- 1/r' must not only be constant for each of
the two bounding surfaces, but must also be equal for
both. Owing to the opposite curvatures of the sur-
faces, then, 1/r -|- 1/r' = 0. Consequently, r = — r'
for all points. Such a surface is called a minimal sur-
face ; that is, it has the smallest area consistent with
its containing certain closed contours. It is also a sur-
face of zero-sum of principal curvatures ; and its ele-
ments, as we readily see, are saddle-shaped. Surfaces
of this kind are obtained by constructing closed space-
curves of wire and dipping the wire into a solution of
soap and water.* The soap-film assumes of its own
accord the form of the curve mentioned.

6. Liquid figures of equilibrium, made up of thin Plateau's
films, possess a peculiar property. The work of the uresofequi-
forces of gravity affects the entire mass of a liquid ;
that of the molecular forces is restricted to its super-
ficial film. Generally, the work of the forces of grav-
ity preponderates. But in thin films the molecular
forces come into very favorable conditions, and it is
possible to produce the figures in question without
difficulty in the open air. Plateau obtained them by
dipping wire polyhedrons into solutions of soap and
water. Plane liquid films are thus formed, which meet

* The mathematical problem of determining such a surface, when the
forms of the wires are given, is called Piattau's Problem, — TVans.

394 THE SCIENCE OF MECHANICS. .

one another at the edges of the framework. When
thin plane films are so joined that they meet at a hol-
low edge, the law 1/r + 1/r' = const no longer holds
for the liquid surface, as this sum has the value zero
for plane surfaces and for the hollow edge a very large
negative value. Conformably, therefore, to the views
above reached, the liquid should run out of the films,
the thickness of which would constantly decrease, and
escape at the edges. This is, in fact, what happens.
But when the thickness of the films has decreased to a
certain point, then, for physical reasons, which are, as
it appears, not yet perfectly known, a state of equilib-
rium is effected.

Yet, notwithstanding the fact that the fundamental
equation l/r+l/^' = const is not fulfilled in these fig-
ures, because very' thin liquid films, especially films of
viscous liquids, present physical conditions somewhat
different from those on which our original suppositions
were based, these figures present, nevertheless, in all
cases a minimum of superficial area. The liquid films,
connected with the wire edges and with one another,
always meet at the edges by threes at approximately
equal angles of 120°, and by fours in comers at approxi-
mately equal angles. And it is geometrically demon-
strable that these relations correspond to a minimum
of superficial area. In the great diversity of phenom-
ena here discussed but one fact is expressed, namely
that the molecular forces do work, positive work, when
the superficial area is diminished.
The reason 7- The figurcs of equilibrium which Plateau ob-

the forms of.»jij. •_ • ii_j • i^* t

equilibrium tamed by dippmg wire polyhedrons m solutions of
metrical, soap, form Systems of liquid films presenting a re-
markable symmetry. The question accordingly forces
itself upon us. What has equilibrium to do with sym-

THE EXTENSION OF THE PRINCIPLES.

395

metry and regularity ? The explanation is obvious.
In every symmetrical system every deformation that
tends to destroy the symmetry is complemented by an
equal and opposite deformation that tends to restore it.
In each deformation positive or negative work is done.
One condition, therefore, though not an absolutely
sufficient one, that a maximum or minimum of work
corresponds to the form of equilibrium, is thus sup-
plied by symmetry. Regularity is successive symme-
try. There is no reason, therefore, to be astonished
that the forms of equilibrium are often symmetrical
and regular.

. 8. The science of mathematical hydrostatics arose The fiRTire
in connection with a special problem — that of the figure

FiR. 2o6.

of the earth. Physical and astronomical data had led
Newton and Huygens to the view that the earth is an
oblate ellipsoid of revolution. Newton attempted to
calculate this oblateness by conceiving the rotating
earth as a fluid mass, and assuming that all fluid fila-
ments drawn from the surface to the centre exert the
same pressure on the centre. Huvgens's assumption
was that the directions of the forces are perpendicular
to the superficial elements. Bouguer combined both
assumptions. Clairauf, finally {Theorie de la figure
de la terre, Paris, 1743), pointed out that the fulfilment
of both conditions does not assure the subsistence of
equilibrium.

396

THE SCIENCE OF MECHANICS.

Clairaut's
point of
view.

Conditions
of equilib-
riMm of
Clairaut's
canals.

Clairaut's starting-point is this. If the fluid earth
is in equilibrium, we may, without disturbing its equi-
librium, imagine any portion of it solidified. Accord-
ingly, let all of it be solidified but a canal ABy of any
form. The liquid in this canal must also be in equilib-
rium. But now the conditions which control equilib-
rium are more easily investigated. If equilibrium exists
in et^ery imaginable canal of this kind, then the entire
mass will be in equilibrium. Incidentally Clairaut re-
marks, that the Newtonian assumption is realised when
the canal passes through the centre (illustrated in Fig.
206, cut 2), and the Huygenian when the canal passes
along the surface (Fig. 206, cut 3).

But the kernel of the problem, according to Clai-
raut, lies in a different view. In all imaginable canals,

Fig. 207. Fig. ao8.

even in one which returns into itself, the fluid must be
in equilibrium. Hence, if cross-sections be made at
any two points M and N of the canal of Fig. 207, the
two fluid columns MPN and MQN must exert on the
surfaces of section at J/ and A^ equal pressures. The
terminal pressure of a fluid column of any such canal
cannot, therefore, depend on the length and the form
of the fluid column, but must depend solely on the po-
sition of its terminal points.

Imagine in the fluid in question a canal MN of any
form (Fig. 208) referred to a system of rectangular co-

THE EXTENSION OF THE PRINCIPLES. 397

ordinates. Let the fluid have the constant density p Maihemai-
and let the force-components X^ V, Z acting on unit of sion of
mass of the fluid in the coordinate directions, be f unc- ditions, and
tions of the codrdinates x, y, z of this mass. Let the quent Ren-
element of length of the canal be called dSf and let its tion of

, 1 1 , » mi e liquid equi-

projections on the axes be ax, ay, dz. The force-com- libnum.
ponents acting on unit of mass in the direction of the
canal are then X(dx/ds), Y{dy/ds), Z(jiz/ds). Let
q be the cross-section ; then, the total force impelling
the element of mass pqds in the direction ds, is

This force must be balanced by the increment of pres-
sure through the element of length, and consequently
must be put equal to q . dp. We obtain, accordingly,
dp=i p {Xdx + Ydy -|- Zdzy The difference of pres-
sure (/) between the two extremities M and N is found
by integrating this expression from Mto N, But as this
difference is not dependent on the form of the canal
but solely on the position of the extremities M and iV,
it follows that p^Xdx-^- Ydy-^- Zdz), or, the density
being constant, Xdx + Ydy + Zdz, must be a com-
plete differential. For this it is necessary that

y^dCr ^_dU dU

^-di' dy' ^^dz'

where 6^ is a function of coordinates. Hence, according
to Clairaut, the general condition of liquid equilibrium is,
that the liquid be controlled by forces which can be ex-
pressed as the partial differential coefficients of one and
the same function of coordinates,

9. The Newtonian forces of gravity, and in fact all
central forces, — forces that masses exert in the direc-
tions of their lines of junction and which are functions

398 THE SCIENCE OF MECHANICS,

Character of the distances between these masses, — possess this
forces property. Under the action of forces of this character
produce the equilibrium of fluids is possible. If we know Uy

equilibrium « i /• • ,

we may replace the first equation by

dp = pdU and / = p^+ const.

The totality of all the points for which 1/= const
is a surface, a so-called level surface. For this surface
also / = const. As all the force- relations, and, as we
how see, all the pressure-relations, are determined by
the nature of the function 6^, the pressure-relations,
accordingly, supply a diagram of the force-relations,
as was before remarked in page 98.
ciairaut's In the theory of Clairaut, here presented, is con-

eerni of the tained, bcyond all doubt, the idea that underlies the
potential, doctrinc of force-function or potential, which was after-
wards developed with such splendid results by La-
place, Poisson, Green, Gauss, and others. As soon
as our attention has been directed to this property of
certain forces, namely, that they can be expressed as
derivatives of the same function U, it is at once recog-
nised as a highly convenient and economical course to
investigate in the place of the forces themselves the
function U.

If the equation

dp = p {Xdx -+- Ydy + Zdz) = pdU

be examined, it will be seen that A'«& + Ydy -\- Zdz
is the element of the work performed by the forces on
unit of mass of the fluid in the displacement ds, whose
projections are dx, dy, dz. Consequently, if we trans-
port unit mass from a point for which U^= C^ to an-

THE EXTENSION OF THE PRINCIPLES. 399

Other point, indifferently chosen, for which U = C,, charactei^

r rr y-. i istics of the

or, more generally, from the surface u=^C^ to the force-func-
surface C^= C^, we perform, no matter by what path
the conveyance has been effected, the same amount of
work. All the points of the first surface present, with
respect to those of the second, the same difference of
pressure j the relation always being such, that

where the quantities designated by the same indices •
belong to the same surface.

10. Let us picture to ourselves a group of such character-
very closely adjacent surfaces, of which every two sue- level, or
cessive ones differ from each other by the same, very tiai, sur-

faces.

small, amount of work required to transfer a mass from
one to the other ; in other words, imagine the surfaces
C/= C, U= C+ dC, U= C+2dC, and so forth.

A mass moving on a level surface evidently per-
forms no work. Hence, every component force in a
direction tangential to the
surface is = 0 ; and the di-
rection of the resultant
force IS everywhere normal
to the surface. If we call dn
the element of the normal
intercepted between two
consecutive surfaces, andy
the force requisite to con-
vey unit mass from the
one surface to the other
through this element, the

work done is/, dn =zd C. As dC is by hypothesis every-
where constant, the force /= dC/dn is inversely pro-
portional to the distance between the surfaces consid-

400 THE SCIENCE OF MECHANICS.

ered. If, therefore, the surfaces U are known, the
directions of the forces are given by the elements of a
system of curves everywhere at right angles to these
surfaces, and the inverse distances between the sur-
faces measure the magnitude of the forces. * These sur-
faces and curves also confront us in the other depart-
ments of physics. We meet them as equi potential
surfaces and lines of force in electrostatics and mag-
netism, as isothermal surfaces and lines of flow in the
theory of the conduction of heat, and as equipotential
surfaces and lines of flow in the treatment of electrical
and liquid currents,
ninstration II. We will uow illustrate the fundamental idea of
ram's doc- Clairaut*s doctrine by another, very simple example.

trine by a _ . ,, i- i t i

simple Imagme two mutually perpendicular planes to cut the
paper at right angles in the straight lines OX and O Y
(Fig. 210). We assume that a force-function exists
6^= — xy^ where x and 7 are the distances from the
two planes. The force-components parallel to OX and
dPKare then respectively

and

y dU

y = -— = — X.

* The same conclasion may be reached as follows. Imagine a water pipe
laid from New York to Key West, with its ends turning up vertically, and of
glass. Let a quantity of water be poured into it, and when equilibrium is
attained, let its height be marked on the glass at both ends. These two marks
will be on one level surface. Now pour in a little more water and again mark
the heights at both ends. The additional water in New York balances the
additional water in Key West. The gravity of the two are equal. But their
quantities are proportional to the vertical distances between the marks.
Hence, the f<;^rce of gravity on a fixed quantity of water is inversely as those
vertical distances, that is, inversely as the distances between consecutive
level surfaces.— TVamx

THE EXTENSION OF THE PRINCIPLES.

401

The level surfaces are cylindrical surfaces, whose
generating lines are at right angles to the plane of the
paper, and whose directrices, xy = const, are equi-
lateral hyperbolas. The lines of force are obtained by
turning the first mentioned system of curves through
an angle of 45° in the plane of the paper about O, If
a unit of mass pass
from the point rtoO ^

by the route rpO, or
r^O, or by any other
route, the work done
is always Op Y. Oq.
If we imagine a
closed canal OprqO
filled with a liquid,
the liquid in the ca-
nal will be in equi-
librium. If transverse
sections be made at
any two pomts, each

section will sustain at both its surfaces the same
pressure.

We will now modify the example slightly. Let the a modifica
forces be A'= — 7, K= — «, where a has a constant example,
value. There exists now no function U so constituted
that X^ dUjdx and K= dUjdy ; for in such a case it
would be necessary that dX/dy = d Y/dx, which is ob-
viously not true. There is therefore no force- function,
and consequently no level surfaces. If unit of mass
be transported from r to Ohy the way of /, the work
done is « X Oq. If the transportation be effected by
the route rqO, the work done is « X Oq ■\- Op y, Oq.
If the canal OprqO were filled with a liquid, the liquid
could not be 'in equilibrium, but would be forced to

402 THE SCIENCE OF MECHANICS.

rotate constantly in the direction OprqO, Currents of
this character, which revert into themselves but con-
tinue their motion indefinitely, strike us as something
quite foreign to our experience. Our attention, how-
ever, is directed by this to an important property of
the forces of nature, to the property, namely, that the
work of such forces may be expressed as a function of
coordinates. Whenever exceptions to this principle
are observed, we are disposed to regard them as appa-
rent, and seek to clear up the difficulties involved.
Torricciii'8 12. We shall now examine a few problems of liquid
on the veio- motion. The founder of the theory of hydrodynamics is
quid efflui. ToRRiCELLi. Torricelli,* by observations on liquids dis-
charged through orifices in the bottom of vessels, dis-
covered the following law. If the time occupied in the
complete discharge of a vessel be divided into n equal
intervals, and the quantity discharged in the last, the
«'*», interval be taken as the unit, there will be dis-
charged in the (« — 1)^, the (« — 2y*>, the(« — 3)^**

interval, respectively, the quantities 3, 5, 7 ... . and
so forth. An analogy between the motion of falling
bodies and the motion of liquids is thus clearly sug-
gested. Further, the perception is an immediate one,
that the most curious consequences would ensue if the
liquid, by its reversed velocity of efflux, could rise
higher than its original level. Torricelli remarked,
in fact, that it can rise at the utmost to this height,
and assumed that it would rise exactly as high if all
resistances could be removed. Hence, neglecting all
resistances, the velocity of efflux, Vy of a liquid dis-
charged through an orifice in the bottom of a vessel is
connected with the height h of the surface of the liquid
by the equation r = 1^2^// ; that is to say, the velocity

* Dt Motn Cravium Proftctorum^ 1643.

THE EXTENSION OF THE PRINCIPLES, 403

of efflux is the final velocity of a body freely falling
through the height h^ or liquid-head ; for only with
this velocity can the liquid just rise again to the sur-
face.*

Torricelli's theorem consorts excellently with the»varignon'«

deduction

rest of our knowledge of natural processes ; but we of the veio-
feel, nevertheless, the need of a more exact insight. eMux.
Varignon attempted to deduce the principle from the
relation between force and the momentum generated by
force. The familiar equation pi =:mv gives, if by a
we designate the area of the basal orifice, by h the
pressure-head of the liquid, by s its specific gravity,
by g the acceleration of a freely falling body, by v the
velocity of efflux, and by t a small interval of time,
this result

ahs . T :^ . V or v^ =^zh,

Here ahs represents the pressure acting during the

time T on the liquid mass avrs/g. Remembering that

z^ is a final velocity, we get, more exactly,

, a - . TS

ahs . r = 2 .Vy

g ~
and thence the correct formula

v^ =2gh,

13. Daniel Bernoulli investigated the motions of
fluids by the principle of vis viva. We w?ll now treat
the preceding case from this point of view, only ren-
dering the idea more modern. The equation which we
employ is/j = mv^ Ji, In a vessel of transverse sec-
tion q (Fig. 211), into which a liquid of the specific

* The early inquirers deduce their propositions in the incomplete form of
proportions, and therefore usually put v proportional to ^gk or ^X.

4<M

THE SCIENCE OF MECHAXICS,

Daniel Ber- gravity s is poured till the head h is reached, the surface

treatment sinks, say, the small distance dh, and the liquid mass

problem, q. dh, s/g IS discharged with the velocity v» The work

done is the same as though the weight q, dh,s had

descended the distance h. The path of the motion in

the vessel is not of consequence here. It makes no

difference whether the stratum q . dh
is discharged directly through the
basal orifice, or passes, say, to a
position a, while the liquid at a is
displaced to ^, that at b displaced to
c, and that at c discharged. The work
done is in each case q , dh . s . h.
Equating this work to the vis viva of the discharged
liquid, we get

Fig. 311.

q , dh . s , h=z~

dh
g

s v^

v = V'2,gh,

The sole assumption of this argument is that all

the work done in the vessel appears as vis viva in the

liquid discharged, that is to say, that the velocities

within the vessel and the work spent in overcoming

friction therein may be neglected. This assumption is

not very far from the truth if vessels of sufficient width

are employed, and no violent rotatory motion is set up.

The law of Let US ueglect the gravity of the liquid in the ves-

w%en pro^* sel, and imagine it loaded by a movable piston, on

th*e^pres^ whosc surface-unit the pressure / falls. If the piston

fill re of

pistons. be displaced a distance dh, the liquid volume q . dh
will be discharged. Denoting the density of the liquid
by p and its velocity by Vy we then shall have

q.p.dh=:q.dh,p ^^ , or 7' = ^

THE EXTENSION OF THE PRINCIPLES. 405

Wherefore, under the same pressure, different liquids
are discharged with velocities inversely proportional to
the square root of their density. It is generally sup>
posed that this theorem is directly applicable to gases.
Its form^ indeed, is correct ; but the deduction fre-
quently employed involves an error, which we shall
now expose.

14. Two vessels (Fig. 212) of equal cross-sections The appii-

^ V o / ^ cation of

are placed side by side and connected with each other this last re

salt to the

by a small aperture in the base of their dividing walls, flow of
For the velocity of flow through this aperture we ob-
tain, under the same suppositions as before,

If we neglect the gravity of the liquid and imagine
the pressures p^ and /^ produced by pistons, we shall
similarly have v=iV 2(/, — P'^lP- For example, if the
pistons employed be loaded with the weights P and
7^/2, the weight P will sink the distance h and PJT,
will rise the distance //. The work (^P/2)h is thus left,
to generate the vis viva of the effluent fluid.

A gas under such circumstances would behave dif- The behav-
ferently. Supposing the gas to flow from the vessel eas under
containing the load /'into that contain- snmed con-

ing the load P/2, the first weight will ^"L— J^^ T
fall a distance //, the second, however, px?| I
since under half the pressure a gas dou- Ep^SppM^
bles its volume, will rise a distance 2 //, p— A ~^
so that the work Ph — (/'/2) 2^ = 0 E^:?! ~ J
would be performed. In the case of „.

* ^ Fig. 212.

gases, accordingly, some additional
work, competent to produce the flow between the vessels
must be performed. This work the gas itself performs,
by expanding, and by overcoming by its force of expan-

406 THE SCIENCE OF MECHANICS,

The result siofi a pressurc. The expansive force / and the volume
form but w oi 9l gas Stand to each other in the familiar relation

differentia .1.1 t ^ «

maKoitude./ttf =>e, where ki SO long as the temperature of the
gas remains unchanged, is a constant. Supposing the
volume of the gas to expand under the pressure / by
an amount dw^ the work done is

For an expansion from w^ to w, or for an increase of
pressure from /^ to /, we get for the work

Conceiving by this work a volume of gas w^ of
density p, moved with the velocity i>, we obtain

/2/„ log

V p

The velocity of efflux is, accordingly, in this case also
inversely proportional to the square root of the density ;
Its magnitude, however, is not the same as in the case
of a liquid,
incom- But even this last view is very defective. Rapid

this view, changes of the volumes of gases are always accom-
panied with changes of temperature, and, consequently
also with changes of expansive force. For this reason,
questions concerning the motion of gases cannot be
dealt with as questions of pure mechanics, but always
involve questions of heat, [Nor can even a thermo-
dynamical treatment always suffice : it is sometimes
necessary to go back to the consideration of molecular
motions.]

15. The knowledge that a compressed gas contains
stored- up work, naturally suggests the inquiry, whether

THE EXTENSION OF THE PRINCIPLES. 40J

this is not also true of compressed liquids. As a mat- Relative

volumes of

ter of fact, every liquid under pressure s's compressed, compressed

ffsses and

To effect compression work is requisite, which reap- fiquids.
pears the moment the liquid expands. But this work,
in the case of the mobile liquids, is very small. Imag-
ine, in Fig. 213, a gas and a mobile liquid of the same
volume, measured by OA, subjected to the same pres-
sure, a pressure of one atmosphere, designated b}' AB.
If the pressure be reduced to one-half an atmosphere,
the volume of the gas will be doubled, while that of
the liquid will be increased by only about 25 millionths.
The expansive work of the gas is represented by the
surface ABDC, that of the liquid by ABLK^ where

Fig. 213.

AJC=0'00002^0A. If the pressure decrease till it
become zero, the total work of the liquid is represented
by the surface AB/, where A/= ooooo^OA, and the
total work of the gas by the surface contained between
AB, the infinite straight line ACEG . . . ., and the
infinite hyperbola branch BDFH . . . . Ordinarily,
therefore, the work of expansion of liquids may be
neglected. There are however phenomena, for ex-
ample, the soniferous vibrations of liquids, in which
work of this very order plays a principal part. In such
cases, the changes of temperature the liquids undergo
must also be considered. We thus see that it is only
by a fortunate concatenation of circumstances that we
are at liberty to consider a phenomenon with any close

4o8

THE SCIENCE OF MECHANICS,

approximation to the truth as a mere matter of molar
mechanics.
The hydro- 1 6. We now come to the idea which Daniel Ber-

dynamic

principle NOULLi sought to apply in his work Hydrodynamica^ sive

of Daniel

Bernoulli, de Viribus et Motibus Fluidorutn Commentarii {iT^B^).
When a liquid sinks, the space through which its cen-
tre of gravity actually descends {descensus actualis) is
equal to the space through which the centre of gravity
of the separated parts affected with the velocities ac-
quired in the fall can ascend {ascensus potentialis). This
idea, we see at once, is identical with that employed
by Huygens. Imagine a vessel filled with a liquid

(Fig. 214) ; and let its horizontal cross-
section at the distance x from the plane
of the basal orifice, be called /(.x). Let
the liquid move and its surface descend
a distance dx. The centre of gravity,
then, descends the distance x/{x) . dx/M,
where M= f/(,x) dx. If k is the space of
potential ascent of the liquid in a cross-
section equal to unity, the space of po-
tential ascent in the cross-section /(:r) will heJf://(x)^,
and the space of potential ascent of the centre of

/fjcj

-"/dx

^-?— ?

.-rJ

-~- -\

f- — — -

_ .. 1 1

m- - —

- — — —%

r — — ■

- — . f

%~ ~ ~

- _ _ -M

— - . J

^fc~ — ,

_ _ -^

%- -

_ _ W

%-~

Fig. ai4.

gravity will be

= k

where

For the displacement of the liquid's surface through a
distance dx, we get, by the principle assumed, both
iVand k changing, the equation

— xf{x) dx = Ndk + kdN.

THE EXTENSION OF THE PRINCIPLES,

409

This equation was employed by Bernoulli in the solu- The parai-
tion of various problems. It will be easily seen, that strata.
Bernoulli's principle can be employed with success
only when the relative velocities of the single parts of
the liquid are known. Bernoulli assumes, — an assump-
tion apparent in the formulae, — that all particles once
situated in a horizontal plane, continue their motion
in a horizontal plane, and that the velocities in the
different horizontal planes are to each other in the in-
verse ratio of the sections of the planes. This is the
assumption of the parallelism of strata. It does not, in
many cases, agree with the facts, and in others its
agreement is incidental. When the vessel as compared
with the orifice of efHux is very wide, no assumption
concerning the motions within the vessel is necessary,
as we saw in the development of Torricelli's theorem.

17. A few isolated cases of liquid motion were The water-

- , __ , _ __ ___ , ,, pendnlum

treated by Newton and John Bernoulli. We shall of Newton,
consider here one to which a
familiar law is directly applic-
able. A cylindrical U-tube with
vertical branches is filled with
a liquid (Fig. 215). The length
of the entire liquid column is /.
If in one of the branches the
column be forced a distance x
below the level, the column in

the other branch will rise the distance x, and the
difference of level corresponding to the excursion x
will be 2 jr. If or is the transverse section of the tube
and s the liquid's specific gravity, the force brought
into play when the excursion x is made, will be ^asx,
which, since it must move a mass otls/gviiW determine
the acceleration (2 asx)/{als/g) = {2g/l) x, or, for unit

Fig. 215.

Bernoulli.

410 THE SCIENCE OF MECHANICS,

excursion, the acceleration 2^//. We perceive that
pendulum vibrations of the duration

will take place. The liquid column, accordingly, vi-
brates the same as a simple pendulum of half the length
of the column.
The liquid A similar, but somewhat more general, problem was

pendulum

of John treated by John Bernoulli. The two branches of a
cylindrical tube (Fig. 216), curved in any manner, make

with the horizon, at the
points at which the
surfaces of the liquid
move, the angles a
and p. Displacing one
of the surfaces the dis-
tance x^ the other sur-
face suffers an equal
displacement. A difference of level is thus produced
X (sin a -\- sin/?), and we obtain, by a course of reason-
ing similar to that of the preceding case, employing
the same symbols, the formula

Fig. 2x6.

= "\l.

(^ (sin or -f sin/^) *

The laws of the pendulum hold true exactly for the
liquid pendulum of Fig. 215 (viscosity neglected), even
for vibrations of great amplitude ; while for the filar
pendulum the law holds only approximately true for
small excursions.

18. The centre of gravity of a liquid as a whole can
rise only as high as it would have to fall to produce its
velocities. In every case in which this principle appears
to present an exception, it can be shown that the excep-

THE EXTENSION OF THE PRINCIPLES. 411

tion is only apparent. One example Js Hero's fountain. Hewi

This apparatus, as we know, consists of three vessels,

which may be designated in the descending order as

A, B, C. The water in the open vessel A

falls through a tubt

C ; the air displace'

on the water in th

this pressure force

jet above A wheni

original level. Thi

true, considerably

but in actuality it

circuitous route of

vessel A to the mu

Another ap- b*"'"?,'''*

parent exception dnoiic

, to the principle .
in question is
that of Montgol-
fier's hydraulic
ram,in which the
liquid by its own
gravitational
work appears to
rise considerably
above its original
level. The liquid
flows (Fig. 217)

from a cistern A p,

through a long

pipe RR and a valve V, which opens inwards, into a
vessel B. When the current becomes rapid enough, the
valve V is forced shut, and a liquid mass m affected with
the velocity ?' is suddenly arrested in RR, which must

412

THE SCIENCE OF MECHANICS,

be deprived of its momentum. If this be done in the
time /, the liquid can exert during this time a pressure
qz=imv/tj to which must be added its hydrostatical
pressure p. The liquid, therefore, will be able, during
this interval of time, to penetrate with a pressure p -{- q
through a second valve into a pila HeroniSy H^ and in
consequence of the circumstances there existing will
rise to a higher level in the ascension-tube 55 than
that corresponding to its simple pressure /. It is
to be observed here, that a considerable portion of the
liquid must first flow off into By before a velocity requi-
site to close V\s produced by the liquid's work in RR,
A small portion only rises above the original level \
the greater portion flows from A into B, If the liquid
discharged from 55 were collected, it could be easily
proved that the centre of gravity of the quantity thus
discharged and of that received in B lay, as the result
of various losses, actually below the level of A,

The principle of the hydraulic ram, that of the

An illastra-

elucidates transference of work done by a large liquid mass to a

the action
of the hy-
draulic ram

smaller one, which
thus acquires a great
vis viva^ may be illus-
trated in the following
very simple manner.
Close the narrow
opening (7 of a funnel
and plunge it, with its
wide opening down-
wards, deep into a
large vessel of water. If the finger closing the upper
opening be quickly removed, the space inside the
funnel will rapidly fill with water, and the surface of the
water outside the funnel will sink. The work performed

Fig. ai8.

THE EXTENSION OF THE PRINCIPLES. 413

is equivalent to the descent of the contents of the funnel
from the centre of gravity S of the superficial stratum
to the centre of gravity *S" of the contents of the fun-
nel. If the vessel is sufficiently wide the velocities in
it are all very small, and almost the entire vis viva is
concentrated in the contents of the funnel. If all the
parts of the contents had the same velocities, they
could all rise to the original level, or the mass as a
whole could rise to the height at which its centre of
gravity was coincident with S, But in the narrower
sections of the funnel the velocity of the parts is
greater than in the wider sections, and the former
therefore contain by far the greater part of the vis
viva. Consequently, the liquid parts above are vio-
lently separated from the parts below and thrown
out through the neck of the funnel high above the
original surface. The remainder, however, are left
considerably below that point, and the centre of grav-
ity of the whole never as much as reaches the original
level of 5.

19. One of the most important achievements of Hydrostatic
Daniel Bernoulli is his distinction of hydrostatic and dynamic

pressure.

hydrodynamic pressure. The pressure

which liquids exert is altered by motion ;

and the pressure of a liquid in motion

may, according to the circumstances, be

greater or less than that of the liquid at rest

with the same arrangement of parts. We

will illustrate this by a simple example.

The vessel -4, which has the form of a body

of revolution with vertical axis, is kept

constantly filled with a frictionless liquid, so that its

surface 2X mn does not change during the discharge

at kl. We will reckon the vertical distance of a particle

414 THE SCIENCE OF MECHANICS.

Detcnnina- from the surf ace m n downwards as positive and call

tion of the . rtii t •••! <•

pressures it s. Let US follow the course of a prismatic element of
acting in li- volume, whose horizontal base- area is a and height /?,
motion. in its downward motion, neglecting, on the assump-
tion of the parallelism of strata, all velocities at right
angles to z. Let the density of the liquid be /}, the
velocity of the element v, and the pressure, which is
dependent on 0, p. If the particle descend the dis-
tance dzy we have by the principle of vis viva

aftpd\-^\ = afipgdz — aj~fidz (1)

that is, the increase of the vis viva of the element is
equal to the work of gravity for the displacement in
question, less the work of the forces of pressure of the
liquid. The pressure on the upper surface of the element
is apj that on the lower surface is a[/ -[" C^//^^)/^]*
The element sustains, therefore, if the pressure in-
crease downwards, an upward pressure a {dp/dz)ft ;
and for any displacement dz of the element, the work
a{dp/dz^ftdz must be deducted. Reduced, equation
(i) assumes the form

and, integrated, gives

„ ^ pgz — / -[- const (2)

If we express the velocities in two different hori-
zontal cross-sections a^ and a^ at the depths s, and z^
below the surface, by v^, v^, and the corresponding
pressures by p^, p^^ we may write equation (2) in the
form

THE EXTENSION OF THE PRINCIPLES. 415
Taking for our cross-section a^ the surface, z. = 0, The hydro-

dynamic

^ J = 0 ; and as the same quantity of liquid flows through pressure
all cross-sections in the same interval of time, a. v. =thecircum-

stances of

^2 ^2' Whence, finally, the motion.

/2 = 9S^

The pressure p^ of the liquid in motion (the hydro-
dynamic pressure) consists of the pressure pgz^ of the
liquid at rest (the hydrostatic pressure) and of a pres-
sure (/>/2)z'5 [(fl| — ^X)I^X\ dependent on the density,
the velocity of flow, and the cross-sectional areas. In
cross-sections larger than the surface of the liquid, the
hydrodynamic pressure is greater than the hydrostatic,
and vice versa,

A clearer idea of the significance of Bernoulli's illustration

. .of these re-

principle may be obtained by imagining the liquid in suits by the

, , now of li-

the vessel A unacted on by gravity, and its outflow quids under

pressures
>duced
pistons.

produced by a constant pressure /^ on the surface, produced
Equation (3) then takes the form

If we follow the course of a particle thus moving, it
will be found that to every increase of the velocity of
flow (in the narrower cross-sections) a decrease of
pressure corresponds, and to every decrease of the. ve-
locity of flow (in the wider cross-sections) an increase
of pressure. This, indeed, is evident, wholly aside
from mathematical considerations. In the present case
every change oi the velocity of a liquid element must be
exclusively produced by the work of the liquid's forces
of pressure. When, therefore, an element enters into
a narrower cross-section, in which a greater velocity
of flow prevails, it can acquire this higher velocity only

4i6 THE SCIENCE OF MECHANICS.

on the condition that a greater pressure acts on its rear
surface than on its front surface, that is to say, only
when it moves from points of higher to points of lower
pressure, or when the pressure decreases in the direc-
tion of the motion. If we imagine the pressures in
a wide section and in a succeeding narrower section
to be for a moment equal, the acceleration of the ele-
ments in the narrower section will not take place ; the
elements will not escape fast enough ; they will accumu-
late before the narrower section ; and at the entrance
to it the requisite augmentation of pressure will be im-
mediately produced. The converse case is obvious.
< 20. In dealing with more complicated cases, the
° problems of liquid motion, even though viscosity be

neglected, present great difficulties; and when the
enormous effects of viscosity are taken into account,
anything like a dynamical solution of almost every
problem is out of the question. So much so, that al-
though these investigations were begun by Newton,
we have, up to the present time, only been able to
master a very few of the simplest problems of this class,
and that but imperfectly. We shall content ourselves
with a simple example. If we cause a liquid contained
in a vessel of the pressure-head h to flow, not through
an orifice in its base, but through a long cylindrical
tube fixed in its side (Fig. 220), the velocity of efflux

THE EXTENSION OF THE PRINCIPLES. 417

V yrill be less than that deducible from Torricelli's law,
as a portion of the work is consumed by resistances
due to viscosity and perhaps to friction. We find, in
fact, that z; = 1/2 ^ // J , where ^ j < ^. Expressing by h ^
the r^^r/Vy-head, and by ^3 the resistance-hi&dA, we may
put A = ^, + '^2* ^^^^ the main cylindrical tube we
affix vertical lateral tubes, the liquid will rise in the
latter tubes to the heights at which it equilibrates the
pressures in the main tube, and will thus indicate at all
points the pressures of the main tube. The noticeable
fact here is, that the liquid-height at the point of influx
of the tube is = Z/^, and that it diminishes in the direc-
tion of the point of outflow, by the law of a straight
line, to zero. The elucidation of this phenomenon is
the question now presented.

Gravity here does not act directly on the liquid in The condi-
the horizontal tube, but all effects are transmitted to it perform- ^
by the pressure of the surrounding parts. If we imag-work in

sucli CAses.

ine a prismatic liquid element of basal area a and
length ft to be displaced in the direction of its length
a distance dz, the work done, as in the previous case, is

dz^ ^ dz

For a finite displacement we have

Work is done when the element of volume is displaced
from a place of higher to a place of lower pressure.
The amount of the work done depends on the size of
the element of volume and on the difference of pressure
at the initial and terminal points of the motion, and
not on the length and the form of the path traversed.

4i8

THE SCIENCE OF MECHANICS.

The conse-
quences of
tnese con-
ditions.

If the diminution of pressure were twice as rapid in
one case as in another, the difference of the pressures
on the front and rear surfaces, or xSx'^ force of the work,
would be doubled, but the space through which the
work was done would be halved. The work done would
remain the same, whether done through the space ab
or ac of Fig. 221.

Through every cross-section q of the horizontal tube
the liquid flows with the same velocity v. If, neglect-
ing the differences of velocity in the same cross- section,
we consider a liquid element which exactly fills the
section q and has the length y5, the vis viva qP p{v^ j'l)
of such an element will persist unchanged throughout

its entire course in the tube.
This is possible only provided
the vis viva consumed by friction
is replaced by the work of the
liquid'* s forces of pressure, H ence,
in the direction of the motion
of the element the pressure
must diminish, and for equal distances, to which the
same work of friction corresponds, by equal amounts.
The total work of gravity on a liquid element q ^p
issuing from the vessel, is qfi pgh. Of this the portion
^/5p(z^2/2) is the vis viva of the element discharged
with the velocity v into the mouth of the tube, or, as
V = V2gh^y the portion q^pgh^. The remainder of
the work, therefore, q fipgh^, is consumed in the tube,
if owing to the slowness of the motion we neglect the
losses within the vessel.

If the pressure-heads respectively obtaining in the
vessel, at the mouth, and at the extremity of the tube,
are^, h^, 0, or the pressures are/ = hgp, p^ = A^gp,0,
then by equation (i) of page 417 the work requisite to

Fig. 221.

THE EXTENSION OF THE PRINCIPLES, 419

generate the vis viva of the element discharged into
the mouth of the tube is

and the work transmitted by the pressure of the liquid
to the element traversing the length of the tube, is

qPp2 = ^PsP^2y

or the exact amount consumed in the tube.

Let us assume, for the sake of argument, that the indirect

- , demonstra*

pressure does not decrease from/- at the mouth touonof
zero at the extremity of the tube by the law of a straight Bequenccs.
line, but that the distribution of the pressure is differ-
ent, say, constant throughout the entire tube. The
parts in advance then will at once suffer a loss of ve-
locity from the friction, the parts which follow will
crowd upon them, and there will thus be produced at
the mouth of the tube an augmentation of pressure
conditioning a constant velocity throughout its entire
length. The pressure at the end of the tube can only
be == 0 because the liquid at that point is not prevented
from yielding to any pressure impressed upon it.

If we imagine the liquid to be a mass of smooth a simile
elastic balls, the balls will be most compressed at the which these
bottom of the vessel, they will enter the tube in a state may*b™*°*
of compression, and will gradually lose that state in ceived.
the course of their motion. We leave the further de-
velopment of this simile to the reader.

It is evident, from a previous remark, that the work
stored up in the compression of the liquid itself, is very
small. The motion of the liquid is due to the work of
gravity in the vessel, which by means of the pressure
of the compressed liquid is transmitted to the parts in
the tube.

420 THE SCIENCE OF MECHANICS.

An interesting modiBcatioD of the case just dis-
cussed is obtained by causing the liquid to flow through
. a tube composed of a number of shorter cylindrical
tubes of varying widths. The pressure in the direction
of outflow then diminishes (Fig. 222) more rapidly in
the narrower tubes, in which a greater consumption of
work by friction takes place, than in the wider ones.
We further note, in every passage of the liquid into a

wider tube, that is to a smaller velocity of flow, an in-
crease of pressure (a positive congestion) ; in every
passage into a narrower tube, that is to a greater velo-
city of flow, an abrupt diminulion of pressure (a nega-
tive congestion). The velocity of a liquid element on
which no direct forces act can be diminished or in-
creased only by its passing to points of higher or lower
pressure.

science.

CHAPTER IV.

THE FORMAL DEVELOPMENT OF MECHANICS.

I.
THE ISOPERIMETRICAL PROBLEMS.

I. When the chief facts of a physical science have The formal,
once been fixed by observation, a new period of its suished

11 •• i.». i»t 1 xromthede-

development begins — the deducttt^e, which we treated ductive, de-
in the previous chapter. In this period, the facts are of physical
reproducible in the mind without constant recourse to
observation. Facts of a more general and complex
character are mimicked in thought on the theory that
they are made up of simpler and more familiar obser-
vational elements. But even after we have deduced
from our expressions for the most elementary facts
(the principles) expressions for more common and more
complex facts (the theorems) and have discovered in
all phenomena the same elements, the developmental
process of the science is not yet completed. The de-
ductive development of the science is followed by its
formal development. Here it is sought to put in a clear
compendious form, or system, the facts to be repro-
duced, so that each can be reached and mentally pic-
tured with the least intellectual effort. Into our rules
for the mental reconstruction of facts we strive to in-
corporate the greatest possible uniformity, so that these
rules shall be easy of acquisition. It is to be remarked,
that the three periods distinguished are not sharply

422 THE SCIEP/CE OF MECHANICS.

separated from one another, but that the processes of
development referred to frequently go hand in hand,
although on the whole the order designated is unmis-
takable.
Theisopari- 2. A powerful influence was exerted on the formal
problems, development of mechanics by a particular class of

and ques- ,

tions of mathematical problems, which, at the close of the

maxima

and minima seventeenth and the beginning of the eighteenth cen-
turies, engaged the deepest attention of inquirers.
These problems, the so-called isoperimetrical problems^
will now form the subject of our remarks. Certain
questions of the greatest and least values of quanti-
ties, questions of maxima and minima, were treated by

the Greek mathemati-
cians. Pythagoras is
said to have taught that
the circle, of all plane
figures of a given peri-
meter, has the greatest
area. The idea, too, of a
certain economy in the processes of nature was not
foreign to the ancients. Hero deduced the law of the
reflection of light from the theory that light emitted
from a point A (Fig. 223) and reflected at J/ will travel
to B by the shortest route. Making the plane of the
paper the plane of reflection, SS the intersection of
the reflecting surface, A the point of departure, B the
point of arrival, and M the point of reflection of the
ray of light, it will be seen at once that the line AMB' ^
where B is the reflection of B, is a straight line. The
line AMB* is shorter than the line ANB*, and there-
fore also AMB is shorter than ANB, Pappus held
similar notions concerning organic nature ; he ex-

FORMAL DEVELOPMENT,

423

plained, for example, the form of the cells of the honey-
comb by the bees' efforts to economise in materials.

These ideas fell, at the time of the revival of the The re-
searches of
sciences, on not unfruitful soil. They were first taken Kepler, Fer-

up by Fermat and Roberval, who developed a method Robervai.
applicable to such problems. These inquirers ob-
served,— as Kepler had already done, — that a magni-
tude y which depends on another magnitude at, gen-
erally possesses in the vicinity of its greatest and least
values a peculiar property. Let x (Fig. 224) denote
abscissas and^ ordinates. If, while x increases, y pass
through a maximum value, its increase, or rise, will
be changed into a decrease, or
fall; and if it pass through a
minimum value its fall will be
changed into a rise. The neigh-
boring values of the maximum
or minimum value, consequently,
will lie very near each other, and

the tangents to the curve at the points in question will
generally be parallel to the axis of abscissas. Hence,
to find the maximum or minimum values of a quan-
tity, we seek the parallel tangents of its curve.

The method of tangents may be put in analytical The
form. For example, it is required to cut off from a tangents,
given line a a portion x such that the product of the
two segments x and a — x shall be as great as possible.
Here, the product x{a — ^) must be regarded as the
quantity y dependent on x. At the maximum value of
y any infinitely small variation of x^ say a variation ^,
will produce no change in y. Accordingly, the required
value of X will be found, by putting

x{a — jr) = (jf -|- S){a — X — ^
or

424

THE SCIENCE OF MECHANICS.

ax — x^ =.ax ^ a^ — x^ — x^ — x^ — 5'

0 = tf — 2:c — 5.

mal effect.

As ^ may be made as small as we please, we also get

^ = a — 1x\

whence x = «/2.

In this way, the concrete idea of the method of
tangents may be translated into the language of alge-
bra ; the procedure also contains, as we see, the germ
of the differential calculus.
The refrac- Fermat sought to find for the law of the refraction
Manunf- of light an expression analogous to that of Hero for

law of reflection. He remarked

that light, proceeding from a

point A^ and refracted at a

point My travels to B^ not by

the shortest route, but in the

shortest time. If the path AMB

is performed in the shortest

time, then a neighboring path

ANBy infinitely near the real

path, will be described in the

same time. If we draw from N on AM and from M on

NB the perpendiculars NP and MQ, then the second

route, before refraction, is less than the first route by a

•distance J//'=-A^J/ sin a, but is larger than it after

refraction by the distance NQ = NM^vn §, On the

supposition, therefore, that the velocities in the first

and second media are respectively v^ and v^y the time

required for the path AMB will be a minimum when

Fig. 325.

NM^\n a NMsm p

= 0

FORMAL DEVELOPMENT, 425

v^ sinflr

- = • z^ = ^>
7>3 sin p

where n stands for the index of refraction. Hero's law
of reflection, remarks Leibnitz, is thus a special case
of the law of refraction. For equal velocities {v^ = v^,
the condition of a minimum of time is identical with
the condition of a minimum of space.

Huygens, in his optical investigations, applied and Hnygens's
further perfected the ideas of Fermat, considering, not of Fermafs

r Aftfi AJTC ufifi.

only rectilinear, but also curvilinear motions of light,
in media in which the velocity of the light varied con-
tinuously from place to place. For these, also, he
found that Fermat's law obtained. Accordingly, in all
motions of light, an endeavor, so to speak, to produce
results in a minimum of time appeared to be the funda-
mental tendency.

3. Similar maximal or minimal properties were The prob-
brought out in the study of mechanical phenomena, brachisto-
As we have already noticed, John Bernoulli knew that
a freely suspended chain assumes the form for which
its centre of gravity lies lowest. This idea was, of
course, a simple one for the investigator who first rec-
ognised the general import of the principle of virtual
velocities. Stimulated by these observations, inquir-
ers now began generally to investigate maximal and
minimal characters. The movement received its most '
powerful impulse from a problem propounded by John
Bernoulli, in June, 1696* — the problem of the brachis-
tochrone. In a vertical plane two points are situated,
A and B, It is required to assign in this plane the
curve by which a falling body will travel from A to B
in the shortest time. The problem was very ingeniously

* Ada Ernditarum, Leipsic.

426 THE SCIENCE OF MECHANICS,

solved by John Bernoulli himself ; and solutions were
also supplied by Leibnitz, L*H6pital, Newton, and
James Bernoulli.
johnBcr- The most remarkable solution was John Ber-

noulli'sin- „,,... , , , ,

eeniousso- NOULLis owu. This inquirer remarks that problems

lutionofthe . ,

problem of of this class have already been solved, not for the mo-

the brachis- , ,

tochrone. tion of falling bodies, but for the motion of light. He
accordingly imagines the motion of a falling body re-
placed by the motion of
a ray of light. (Comp.
P- 379) The two points
A and B are supposed
to be fixed in a medium
in which the velocity of
light increases in the
vertical downward direction by the same law as the
velocity of a falling body. The medium is supposed
to be constructed of horizontal layers of downwardly

decreasing density, such that v '=-\ 2gh denotes the
velocity of the light in any layer at the distance h be-
low A, A ray of light which travels from A to B un-
der such conditions will describe this distance in the
shortest time, and simultaneously trace out the curve
of quickest descent.

Calling the angles made by the element of the
curve with the perpendicular, or the normal of the
layers, oty a\ or". . . ., and the respective velocities
V, p', v". . . ., we have

sin a sin a' sin a'* .

or, designating the perpendicular distances below A
by X, the horizontal distances from A by y^ and the arc
of the curve by j.

FORMAL DEVELOPMENT, 427

./ (. / chrone a

"^/ __ jf, cycloid.

whence follows

//>'2 _, ^2 y;2 ^j2 — ^2 7,2 (,/>v2 _p ^^2)

and because z^ = 1/2 ^jc also

//v = //jca I — - , where a = ^ — r;r-

This is the differential equation of a cycloid, or curve
described by a point in the circumference of a circle of
radius r=^a/2 = ij^gk^^ rolling on a straight line.

To find the cycloid that passes through A and By The con-
it is to be noted that all cycloids, inasmuch as they are Jhe cycloid

' ■ ^ between

produced by similar con- t^? K>ven

structions, are similar^ ^\

and that if generated by

the rolling of circles on

AD from the point A as

origin, are also similarly

, . 1 Pig* 3?7*

Situate a with respect to

the point A, Accordingly, we draw through AB a
straight line, and construct any cycloid, cutting the
straight line in B. The radius of the generating
circle is, say, r'. Then the radius of the generating
circle of the cycloid sought is r= r\AB/AB').

This solution of John Bernoulli's, achieved entirely
without a method, the outcome of pure geometrical
fancy and a skilful use of such knowledge as happened
to be at his command, is one of the most remarkable
and beautiful performances in the history of physical
science. John Bernoulli was an aesthetic genius in this
field. His brother James's character was entirely differ-
ent. James was the superior of John in critical power,

428 THE SCIENCE OF MECHANICS.

Compiri- but in originality and imagination was surpassed by the
sdMiiifio latter. James Bernoulli likewise solved this problem,
etlobn and though in less felicitous form. But, on the Other hand,
nouiii. he did not fail to develop, with great thoroughness, a
general ra^xSxoA applicable to such problems. Thus,
in these two brothers we find the two fundamental
traits of high scientific talent separated from one
another, — traits, which in the very greatest natural
inquirers, in Newton, for example, are combined to-
gether. We shall soon see those two tendencies, which
within one bosom might have fought their battles un-
noticed, clashing in open conflict, in the persons of
these two brothers.

■( ?M«..

I»ineiB«- 4, James Bernoulli finds that the chief object of
■narks on research hitherto had been to find the values of a vari-
niiurBof able quantity, for which a second variable quantity,
probiim. which is a function of the first, assumes its greatest or
its least value. The present problem, however, is to find

FORMAL DEVELOPMENT, 429

from among an infinite number of curves one which pos-
sesses a certain maximal or minimal property. This, as
he correctly remarks, is a problem of an entirely dif-
ferent character from the other and demands a new
method. •

The principles that James Bernoulli employed inThepHnci-
the solution of this problem {Acta Eruditorum^ May, ployed in

r 11 James Ber-

1697)* are as follows : nouiu'sso-

(i) If a curve has a certain property of maximum
or minimum, every portion or element of the curve has
the same property.

(2) Just as the infinitely adjacent values of the
maxima or minima of a quantity in the ordinary prob-
lems, for infinitely small changes of the independent
variables, are constant, so also is the quantity here to
be made a maximum or minimum for the curve sought,
for infinitely contiguous curves^ constant.

(3) It is finally assumed, for the case of the brachis-
tochrone, that, the velocity is 7/ = 1/2^//, where h de-
notes the height fallen through.

If we picture to ourselves a very small portion ABCthe essen-

tial fea-

of the curve (Fig. 228), and, imagining a horizontal tures of

ft 1 1 « James Ber-

ime drawn through B, cause uoniii'sso-

, . - . j4\. lution.

the portion taken to pass mto
the infinitely contiguous por-
tion ADC, we shall obtain, by
considerations exactly similar
to those employed in the treat-
ment of Fermat's law, the well-
known relation between the

sines of the angles made by the curve-elements with
the perpendicular and the velocities of descent. In
this deduction the following assumptions are made,

* See also his works, Vol. II. p. 768. «

430

THE SCIENCE OF MECHANICS,

The Pro-
gramma of

(i), that the part^ or element, ABC is brachistochro-
nous, and (2), that ADC is described in the same time
as ABC, Bernoulli's calculation is very prolix ; but
its essential features are obvious, and the problem is
solved * by the above-stated principles.

With the solution of the problem of the brachisto-
jamesBer- chrone, Tames Bernoulli, in accordance with the prac-

noulli, or . •* . . . . ^

theproposi-tice then prevailing among mathematicians, proposed

tion of tlio

general iso- the foUowiug more general " isoperimetrical problem '*:

perimetri"

cai prob- <« Of all isoperimetrical curves (that is, curves of equal
"perimeters or equal lengths) between the same two
"fixed points, to find the curve such that the space
"included (i) by a second curve, each of whose ordi-
" nates is a given function of the corresponding ordi-
"nate or the corresponding arc of the one sought, (2)
" by the ordinates of its extreme points, and (3) by the
" part of the axis of abscissae lying between those ordi-
" nates, shall be a maximum or minimum."

For example. It is required to find the curve BFN^
described on the base BN such, that of all curves of

the same length on BN^
this particular one shall make
the area BZN a minimum,
where PZ=(^PFy, LM =
{LKy^ and so on. Let the
relation between the ordi-
nates of BZN and the cor-
responding ordinates of BFN
be given by the curve BH, To obtain PZ from PF,
draw FGH at right angles to BG, where ^6^ is at right
angles to BN. By hypothesis, then, PZ=i GHy and

* For the details of this solution and for information generally on the his-
tory of this subject, see Woodhouse's Treatise on Isoperimetrical ProbUntt
and the Calculus i^ Variations^ Cambridge, z8io. — Trans,

FORMAL DEVELOPMENT, 431

SO for the other ordinates. Further, we put BPz=y,
PF= X, PZ = a:".

John Bernoulli gave, forthwith, a solution of this John Ber-

. noulli'a so-

problem, in the form lutionof

this prob-

where a is an arbitrary constant. For « = 1,

xdx

/xax
== a

— Va^—x'^,

that is, BFN is a semicircle on ^iV as diameter, and
the area BZN is equal to the area BFN, For this par-
ticular case, the solution, in fact, is correct. But the
general formula is not universally valid.

On the publication of John Bernoulli's solution,
James Bernoulli openly engaged to do three things :
first, to discover his brother's method ; second, to point
out its contradictions and errors ; and, third, to give the
true solution. The jealousy and animosity of the two
brothers culminated, on this occasion, in a violent and
acrimonious controversy, which lasted till James's
death. After James's death, John virtually confessed
his error and adopted the correct method of his brother.

James Bernoulli surmised, and in all probability James Ber-

. noulli's

correctly, that John, misled by the results of his re- criticism of

searches on the catenary and the curve of a sail filled nouiH's so-
lution,
with wind, had again attempted an indirect solution,

imagining ^/^A^ filled with a liquid of variable density

and taking the lowest position of the centre of gravity

as determinative of the curve required. Making the

ordinate /'Z=/, the specific gravity of the liquid in

the ordinate PF-=zx must be pjx^ and similarly in

every other ordinate. The weight of a vertical fila-

432

THE SCIENCE OF MECHANICS,

ment is then / . dyjx^ and its moment with respect to

^iV^is

\ pdy 1 ^
x<^=-^pdy.

ution.

Hence, for the lowest position of the centre of gravity,
i fpdy, or Cfi dy =zBZJV, is a maximum. But the
fact is here overlooked, remarks James Bernoulli, that
with the variation of the curve BFN the weight of the
liquid also is varied. Consequently, in this simple
form the deduction is not admissible.
Thefunda- In the solution which he himself gives. Tames Ber-
principle of noulli once more assumes that the small portion F F,,,

James Ber- '^ '"

nouiii'8 of the curve possesses the prop-

eneral so- «. j 7^

erty which the whole curve pos-
sesses. And then taking the four
successive points F F^ F,, F,,,,
of which the two extreme ones
are fixed, he so varies F, and
Fff that the length of the arc F
Ff Fff F,,f remains unchanged^
which is possible, of course, only by a displacement
of two points. We shall not follow his involved and
unwieldy calculations. The principle of the process is
clearly indicated in our remarks. Retaining the des-
ignations above employed, James Bernoulli, in sub-
stance, states that when

fpdy is a maximum, and when

dy=-^^---ML
\^2ap—p^

fpdy is a miuimum.

Fig. aao.

FORMAL DEVELOPMENT. 433

The dissensions between the two brothers were, we
may admit, greatly to be deplored. Yet the genius of
the one and the profundity of the other have borne, in
the stimulus which Euler and Lagrange received from
their several investigations, splendid fruits.

5. Euler iProblematis Isoperimetrici Solutio Generalise Euier^a

general

Com, Acad, Petr, T. VI, for 1733, published in i738)*ciassifica-
was the first to give a more general method of treating isopcrimet-

. ... . rical prob-

these questions of maxima and minima, or isoperimetri- lems.
cal problems. But even his results were based on
prolix geometrical considerations, and not possessed of
analytical generality. Euler divides problems of this
category, with a clear perception and grasp of their
differences, into the following classes :

(i) Required, of a// curves, that for which a prop-
erty ^ is a maximum or minimum.

(2) Required, of all curves, equally possessing a
property Ay that for which -^ is a maximum or mini-
mum.

(3) Required, of all curves, equally possessing two
properties, A and B, that for which C is a maximum
or minimum. And so on.

A problem of the first class is (Fig. 231) the finding Examples.
of the shortest curve through M and N, A problem of
the second class is the finding of a curve through M
and Ny which, having the given length A^ makes the
area MPN a maximum. A problem of the third class
would be : of all curves of the given length A, which
pass through M^ N and contain the same area
MPN=^B, to find one which describes when rotated
about MN the least surface of revolution. And so on.

* Euler's principal contribations to this subject are contained in three
memoirs, published in the Commentaries ^ Peiertburg^ lot \\xe years 1733, 1736,
and X7<S6, and in the tract Methodtu inveniendi Lineas Curva* Pro^ietate
Meucimi Minimive gaudentet^ Lausanne and Geneva, 1744. — Trans.

434 ^^^^' SCIENCE OF MECHANICS,

We may observe here, that the finding of an abso-
lute maximum or minimum, without collateral condi-
tions, is meaningless. Thus, all the curves of which in

the first example the shortest is sought
possess the common property of pas-
sing through the points M and N.

The solution of problems of the
first class requires the variation of two
elements of the curve or of one point.
This is also sufficient. In problems
of the second class three elements or
Fig. a3i. ^^^ points must be varied ; the reason

being, that the varied portion must
possess in common with the unvaried portion the prop-
erty Ay and, as B is to be made a maximum or mini-
mum, also the property B, that is, must satisfy two con-
ditions. Similarly, the solution of problems of the third
class requires the variation of four elements. And
so on.
The com- The solutiou of a problem of a higher class involves,

of the iao- by implication, the solution of its converse, in all its
cai proper- forms. Thus, in the third class, we vary four elements

ties with *

Eui'ers in- of the curve, SO, that the varied portion of the curve
shall share equally with the original portion the values
A and B and, as C is to be made a maximum or a
minimum, also the value C But the same conditions
must be satisfied, if of all curves possessing equally B
and C that for which ^ is a maximum or minimum is
sought, or of all curves possessing A and C that for
which ^ is a maximum or minimum is sought. Thus
a circle, to take an example from the second class, con-
tains, of all lines of the same length Aj the greatest
area By and the circle, also, of all curves containing
the same area B^ has the shortest length A, As the

FORMAL DEVELOPMENT. 435

condition that the property A shall be possessed in
common or shall be a maximum, is expressed in the
same manner, Euler saw the possibility of reducing the
problems of the higher classes to problems of the first
class. Ify for example^ it is required to find, of all
curves having the common property A^ that which
makes B a maximum, the curve is sought for which
A + ^f^B is a maximum, where m is an arbitrary con-
stant. If on any change of the curve, A + ttiB, for any
value of niy does not change, this is generally possible
only provided the change of A^ considered by itself,
and that of B^ considered by itself, are = 0.

6. Euler was the originator of still another impor- The funda-
tant advance. In treatmg the problem of finding the principle o(

. James Ber-

brachistochrone in a resisting medium, which was in- noniU's
vestigated by Herrmann and him, the existing meth- shown not

, 1 . -r-* 1 1 1 • 1 . to be uni-

ods proved mcompetent. For the brachistochrone m versaiiy
a vacuum, the velocity depends solely on the vertical
height fallen through. The velocity in one portion of
the curve is in no wise dependent on the other por-
tions. In this case, then, we can indeed say, that if
the whole curve is brachistochronous, every element
of it is also brachistochronous. But in a resisting
medium the case is different. The entire length and
form of the preceding path enters into the determina-
tion of the velocity in the element. The whole curve
can be brachistochronous without the separate ele-
ments necessarily exhibiting this property. By con-
siderations of this character, Euler perceived, that the
principle introduced by James Bernoulli did not hold
universally good, but that in cases of the kind referred
to, a more detailed treatment was required.

7. The methodical arrangement and the great num-
ber of the problems solved, gradually led Euler to sub-

436 THE SCIENCE OF MECHANICS,

LagranKe's stantially the same methods that Lagrange afterwards
Ei8to%of developed in a somewhat different form, and which
lus of Vari- now go by the name of the Calculus of Variations, First,
John Bernoulli lighted on an accidental solution of a
problem, by analogy. James Bernoulli developed, for
the solution of such problems, a geometrical method.
Euler generalised the problems and the geometrical
method. And finally, Lagrange, entirely emancipating
himself from the consideration of geometrical figures,
gave an analytical method. Lagrange remarked, that
the increments which functions receive in consequence
of a change in their /^rw are quite analogous to the in-
crements they receive in consequence of a change of
their independent variables. To distinguish the two
species of increments, Lagrange denoted the former
by 6y the latter by d. By the observation of this anal-
ogy Lagrange was enabled to write down at once the
equations which solve problems of maxima and minima.
Of this idea, which has proved itself a very fertile one,
Lagrange never gave a verification ; in fact, did not
even attempt it. His achievement is in every respect
a pectiliar one. He saw, with great economical in-
sight, the foundations which in his judgment were suf-
ficiently secure and serviceable to build upon. But
the acceptance of these fundamental principles them-
selves was vindicated only by its results. Instead of
employing himself on the demonstration of these prin-
ciples, he showed with what success they could be em-
ployed. {Essai d^une nouvelle methode pour determiner
les maxima et minima des formules integrates indefinies,
Misc, Taur, 1762.)

The difficulty which Lagrange's contemporaries and
successors experienced in clearly grasping his idea, is
quite intelligible. Euler sought in vain to clear up the

FORMAL DEVELOPMENT. 437

difierence between a variation and a differential by The mis-
imagining constants contained in the function, with Uons of La-
the change of which the form of the function changed, idea.
The increments of the value of the function arising
from the increments of these constants were regarded
by him as the variations, while the increments of the
function springing from the increments of the indepen-
dent variables were the differentials. The conception
of the Calculus of Variations that springs from such a
view is singularly timid, narrow, and illogical, and does
not compare with that of Lagrange. Even LindelSfs
modern work, so excellent in other respects, is marred
by this defect. The first really competent presenta-
tion of Lagrange's idea is, in our opinion, that of Jel-
LETT.* Jellett appears to have said what Lagrange per-
haps was unable fully to say, perhaps did not deem it
necessary to say.

8. Jellett's view is, in substance, this. Quantities J«"«^'««^-
generally are divisible into constant and variable quan- the pnnci-

. . , . pies of the

titles; the latter being subdivided into independent Caicuins of
and dependent variables, or such as may be arbitrarily
changed, and such whose change depends on the
change of other, independent, variables, in some way
connected with them. The latter are called functions
of the former, and the nature of the relation that con-
nects them is termed the form of the function. Now,
quite analogous to this division of quantities into con-
stant and variable, is the division of the forms of func-
tions into determinate (constant) and indeterminate (vari-
able). If the form of a function, y = <p{x)j is inde-
terminate, or variable, the value of the function y can
change in two ways : (i) by an increment dx of the

• An RUmentary TretUist on tkt CaUnhu ef Variaiiinu, By the Rev.
John Hewitt Jellett. Dublin, 183a

438 THE SCIENCE OF MECHANICS.

independent variable jr, or (2) by a change oiform, by
a passage from (pio <p^. The first change is the dif-
ferential dy, the second, the variation dy. Accord-
ingly,

dy= q}(x -\- dx) — q) (^), and

^y = <P\ W — 9 (^)-

The object The change of value of an indeterminate function

of the cal-

cuius of va- due to a mere change of form involves no problem,

riations il-

lustrated. just as the change of value of an independent variable
involves none. We may assume any change of form
we please, and so produce any change of value we
please. A problem is not presented till the change in
value of a determinate function {F^ of an indetermi-
nate function 9?, due to a change of form of the included
indeterminate function, is required. For example, if
we have a plane curve of the indeterminate ioxxa y=-
q> (x)^ the length of its arc between the abscissae x^
and\^j is

a determinate function of an indeterminate function.
The moment a definite form of curve is fixed upon, the
value of 5 can be given. For any change of form of
the curve, the change in value of the length of the arc,
6S^ is determinable. In the example given, the func-
tion S does not contain the function y directly, but
through its first differential coefficient dy/dx, which is
itself dependent on y. Let u = F{y^ be a determinate
function of an indeterminate function j' = <p{x^ ; then

Su=.F{y+Sy)- F{y) ='^-^-}p Sy.

Again, let u =F{y, dyjdx) be a determinate function

FORMAL DEVELOPMENT. 439

of an indeterminate function, y = <p{x). For a change
of form of (p, the value of y changes by dy and the
value of dy/dx by 6{dy/dx). The corresponding change
in the value of u is

dy "^ dy dx

dy
The expression 6 -/- is obtained by our definition from Expres-

ax siona for

the varia-

dy^ d(^y^6y^ _dy ^ djy {l^Stili'*'"

dx dx dx dx' coeffidenta

Similarly, the following results are found :

^d^y^d^dy d^ y^d^dy
dx^ — ~dx^' d~x^ ^ ~d~x^'

and so forth.

We now proceed to a problem, namely, the de-Aprobieia
termination of the form of the function ^ = ^^(x) that
will render

U=J^Vdx
where

a maximum or minimum ; q> denoting an indetermi-
nate, and F a determinate function. The value of U
may be varied (i) by a change of the limits, x^^ x^.
Outside of the limits, the change of the independent
variables x, as such, does not affect U\ accordingly,
if we regard the limits as fixed, this is the only respect
in which we need attend to x. The only other way
(2) in which the value of U is susceptible of variation

440 THE SCIENCE OF MECHANICS.

is by a change of iki^form oiy=. <p(x). This produces
a change of va/ue in

amounting to

* »

and so forth. The total change in [/, which we shall
call DC/, and to express the maximum-minimum con-
dition put = 0, consists of the differential dC/ and the
variation 617. Accordingly,

I>17= dU'+ 6C/= 0.

Expresaion Denoting by V^dx^ and — ^o^-^o ^^® increments of
variation of U due to the change of the limits, we then have

the func-
tion in •**!

question. DU= V^ dx^ — V^ dx^ + 6 J Vdx =

V^ dx^ —V^dx^ -f J^d r. dx = 0.
But by the principles stated on page 439 we further get

^^=dy^y\dy^dx'^-d--y^dx-^ . . . =

dx dx^

dV dVddy dV d^dy^

^ /y '^'^ /""y d'x^'^ —

dx dx^

For the sake of brevity we put

dy "" ^yy "" ^y^'y "" ''

dx dx^

Then

(J rV//jc=r

FORMAL DEVELOPMENT. 441

^o \ / Qie third

term of the

One difl&culty here is, that not only dy, but also the foPfhfjJi" |
terms ddy/dx^ d^ dy/dx^ .... occur in this equation, ▼aria'^on-
— terms which are dependent on one another, but not
in a directly obvious manner. This drawback can be
removed by successive integration by parts, by means
of the formula

Cudv = uv — Cvdu.

By this method

^» -d-x - di ^y + J -d^ *-^''*' "'^^ ^° °°-

Performing all these integrations between the limits,
we obtain for the condition DU^=^ the expression

0 = V^dx^ - V^ dx^

dP^ , \ {d6y\ /_ </^3 , \ idSy\

+i'.-'5+-),(f),-(-'-t^-+-jA*;

which now contains only 6y under the integral sign.

The terms in the first line of this expression are
independent of any change in the form of the function
and depend solely upon the variation of the limits.

442 THE SCIENCE OF MECHANICS.

The inter- The terms of the two following lines depend on the
the results, change in the form of the function, for the limiting
values of x only ; and the indices i and 2 state that
the actual limiting values are to be put in the place of
the general expressions. The terms of the last line,
finally, depend on the general change in the form of
the function. Collecting all the terms, except those in
the last line, under one designation a^ — a^y and calling
the expression in parentheses in the last line /9, we
have

0 = Of, — «o + r^ • ^y ' ^^'

But this equation can be satisfied only if

a,^a^ = 0 (1)

and

f/3dydx = 0 (2)

For if each of the members were not equal to zero,
each would be determined by the other. But the in-
tegral of an indeterminate function cannot be expressed
in terms of its limiting values only. Assuming, there-
fore, that the equation

j)36y^x = 0,

Theequa- holds generally good, its conditions can be satisfied,
solves the since 6y is throughout arbitrary and its generality of
makes the form cannot be restricted, only by making y5 = 0. By

function in

question a the equation

therefore, the form of the function y = q>(x) that makes
the expression C/'a maximum or minimum is defined.

maximum
or mini-
mum.

FORMAL DEVELOPMENT, 443

Equation (3) was found by Euler. But Lagrange first
showed the application of equation (i), for the deter-
mination of a function by the conditions at its limits.
By equation (3), which it must satisfy, the^r^i of the
function y = <p{x) is generally determined ; but this
equation contains a number of arbitrary constants,
whose values are determined solely by the conditions
at the limits. With respect to notation, J ellett rightly
remarks, that the employment of the symbol ^ in the
first two terms V^Sx^ z=zV^dx^ of equation (i), (the
form used by Lagrange,) is illogical, and he correctly
puts for the increments of the independent variables
the usual S5mibols dx^y dx^,

9. To illustrate the use to which these equations a practical

illustration

may be put, let us seek the form of the function that of the use

; '^ ' of these

makes equations.

1 +

a minimum — the shortest line. Here

All expressions except

vanish in equation (3), and that equation becomes
dP^jdx = 0 ; which means that -P,, and consequently
its only variable, dy/dx, is independent of x. Hence,
dy/dx = a, and y = ax -{• b, where a and b are con-
stants.

The constants a, b are determined by the values of

444 THE SCIENCE OF MECHANICS.

Develop- the limits. If the straight line passes through the

ment of the

illustration, points Xq, y^ and x^y y^y then

^o=""«in (-)

and as ^Xq = dx^ = 0, Sy^ = 6y^ = 0, equation (i)
vanishes. The coefficients 6 (dy/dx), ^ {d^yjdx^), ....
independently vanish. Hence, the values of a and b
are determined by the equations {m) alone.

If the limits x^y x^ only are given, but^^, ^^ are
indeterminate, we have dx^ = //x^ = 0, and equation
(i) takes the form

— T^=z3 (^yi — ^yo) = 0,

Vl + a^

which, since Sy^ and 6y^ are arbitrary, can only be
satisfied if ^ = 0. The straight line is in this case
y=zd, parallel to the axis of abscissae, and as ^ is inde-
terminate, at any distance from it.

It will be noticed, that equation (i) and the sub-
sidiary conditions expressed in equation (»i), with re-
spect to the determination of the constants, generally
complement each other.

=/-'\

• + I'i)' ^'

is to be made a minimum, the integration of the appro-
priate form of (3) will give

r . ^

jr — r X — c

y=-o

e -\- e

If Z is a minimum, then 7.71 Z also is a minimum, and
the curve found will give, by rotation about the axis
of abscissae, the least surface of revolution. Further,

FORMAL DEVELOPMENT. 445

to a minimum of Z the lowest position of the centre of
gravity of a homogeneously heavy curve of this kind
corresponds ; the curve is therefore a catenary. The
determination of the constants Cy c^ is effected by means
of the limiting conditions, as above.

In the treatment of mechanical problems, a dis- variations

* and virtual

tinction is made between the increments of coordmates displace-
ments dis-

that actually take place in time, namely, dx, dy, dz, tinguished.
and the possible displacements ^x^ dy^ dz, considered,
for instance, in the application of the principle of vir-
tual velocities. The latter, as a rule, are not varia-
tions; that is, are not changes of value that spring
from changes in the form of a function. Only when
we consider a mechanical system that is a continuum,
as for example a string, a flexible surface, an elastic
body, or a liquid, are we at liberty to regard 6x, 6y,
dz as indeterminate functions of the coordinates x, y,
z, and are we concerned with variations.

It is not our purpose in this work, to develop math- importance
ematical theories, but simply to treat the purely phys- caius of va-

' '^ -^ jT , . . nations for

ical part of mechanics. But the history of the isopen- mechanics,
metrical problems and of the calculus of variations had
to be touched upon, because these researches have ex-
ercised a very considerable influence on the develop-
ment of mechanics. Our sense of the general prop-
erties of systems, and of properties of maxima and
minima in particular, was much sharpened by these
investigations, and properties of the kind referred to
were subsequently discovered in mechanical systems
with great facility. As a fact, physicists, since La-
grange's time, usually express mechanical principles
in a maximal or minimal form. This predilection
would be unintelligible without a knowledge of the
historical development.

446 THE SCIENCE OF MECHANICS,

II.

THEOLOGICAL. ANIMISTIC, AND MYSTICAL POINTS OF VIEW

IN MECHANICS.

I. If, in entering a parlor in Germany, we happen
to hear something said about some man being very
pious, without having caught the name, we may fancy
that Privy Counsellor X was spoken of, — or Herr von
Y ; we should hardly think of a scientific man of our
acquaintance. It would, however, be a mistake to sup-
pose that the want of cordiality, occasionally rising to
embittered controversy, which has existed in our day
between the scientific and the theological faculties,
always separated them. A glance at the history of
science suffices to prove the contrary.
The con- People talk of the ** conflict" of science and the-

ence and ology, or better of science and the church. It is in
truth a prolific theme. On the one hand, we have the
long catalogue of the sins of the church against pro-
gress, on the other side a ** noble army of martyrs,*'
among them no less distinguished figures than Galileo
and Giordano Bruno. It was only by good luck that
Descartes, pious as he was, escaped the same fate.
These things are the commonplaces of history ; but it
would be a great mistake to suppose that the phrase
** warfare of science" is a correct ■ description of its
general historic attitude toward religion, that the only
repression of intellectual development has come from
priests, and that if their hands had been held off, grow-
ing science would have shot up with stupendous velo-
city. No doubt, external opposition did have to be
fought ; and the battle with it was no child's play.

FORMAL DEVELOPMENT, 447

Nor was any engine too base for the church to handle The struR-

. , sleofscien-

in this struggle. She considered nothing but how to tists with

, their own

conquer ; and no temporal policy ever was conducted precon-

ceiveci

SO selfishly, so unscrupulously, or so cruelly. But in- ideas,
vestigators have had another struggle on their hands,
and by no means an easy one, the struggle with their
own preconceived ideas, and especially with the notion
that philosophy and science must be founded on the-
ology. It was but slowly that this prejudice little by
little was erased.

2. But let the facts speak for themselves, while we Historical

, , examples.

introduce the reader to a few historical personages.

Napier, the inventor of logarithms, an austere Puri-
tan, who lived in the sixteenth century, was, in addi-
tion to his scientific avocations, a zealous theologian.
Napier applied himself to some extremely curious
speculations. He wrote an exegetical commentary on
the Book of Revelation, with propositions and mathe-
matical demonstrations. Proposition XXVI, for ex-
ample, maintains that the pope is the Antichrist ; propo-
sition XXXVI declares that the locusts are the Turks
and Mohammedans ; and so forth.

Blaise Pascal (i 623-1 662), one of the most rounded
geniuses to be found among mathematicians and phys-
icists, was extremely orthodox and ascetical. So deep
were the convictions of his heart, that despite the gen-
tleness of his character, he once openly denounced at
Rouen an instructor in philosophy as a heretic. The
healing of his sister by contact with a relic most seri-
ously impressed him, and he regarded her cure as a
miracle. On these facts taken by themselves it might
be wrong to lay great stress ; for his whole family were
much inclined to religious fanaticism. But there are
plenty of other instances of his religiosity. Such was

448 THE SCIENCE OF MECHANICS,

Pascal. his resolve, — ^which was carried out, too, — to abandon
altogether the pursuits of science and to devote his life
solely to the cause of Christianity. Consolation, he
used to say, he could find nowhere but in the teachings
of Christianity ; and all the wisdom of the w6rld availed
him not a whit. The sincerity of his desire for the
conversion of heretics is shown in his Lettres provin-
dales, where he vigorously declaims against the dread-
ful subtleties that the doctors of the Sorbonne had
devised, expressly to persecute the Jansenists. Very
remarkable is Pascal's correspondence with the theo-
logians of his time ; and a modern reader is not a little
surprised at finding this great "scientist" seriously
discussing in one of his letters whether or not the Devil
was able to work miracles.

otto yon Otto von Gucricke, the inventor of the air-pump,

occupies himself, at the beginning of his book, now
little over two hundred years old, with the miracle of
Joshua, which he seeks to harmonise with the ideas
of Copernicus. In like manner, we find his researches
on the vacuum and the nature of the atmosphere in-
troduced by disquisitions concerning the location of
heaven, the location of hell, and so forth. Although
Guericke really strives to answer these questions as ra-
tionally as he can, still we notice that they give him
considerable trouble,— rquestions, be it remembered,
that to-day the theologians themselves would consider
absurd. Yet Guericke was a man who lived after the
Reformation !

The giant mind of Newton did not disdain to employ
itself on the interpretation of the Apocalypse. On such
subjects it was difficult for a sceptic to converse with
him. When Halley once indulged in a jest concerning
theological questions, he is said to have curtly repulsed

FORMAL DEVELOPMENT. 449

him with the remark : ** I have studied these things ; Newton and

Leibnitz.

you have not I "

We need not tarry by Leibnitz, the inventor of the
best of all possible worlds and of pre-established har-
mony— inventions which Voltaire disposed of in Can-
didff a humorous novel with a deeply philosophical pur-
pose. But everybody knows that Leibnitz was almost
if not quite as much a theologian, as a man of science.

Let us turn, however, to the last century. Euler, in Euier.
his Letters to a German Princess^ deals with theologico-
philosophical problems in the midst of scientific ques-
tions. He speaks of the difficulty involved in explaining
the interaction of body and mind, due to the total
diversity of these two phenomena, — a diversity to his
mind undoubted. The system of occasionalism, devel-
oped by Descartes and his followers, agreeably to which
God executes for every purpose of the soul, (the soul it-
self not being able to do so,) a corresponding movement
of the body, does not quite satisfy him. He derides,
also, and not without humor, the doctrine of pre-
established harmony, according to which perfect agree-
ment was established from the beginning between the
movements of the body and the volitions of the soul, —
although neither is in any way connected with the
other, — just as there is harmony between two different
but like-constructed clocks. He remarks, that in this
view his own body is as foreign to him as that of a
rhinoceros in the midst of Africa, which might just as
well be in pre-established harmony with his soul as
its own. Let us hear his own words. In his day, Latin
was almost universally written. When a German
scholar wished to be especially condescending, he
wrote in French : * ' Si dans le cas d*un d^r^glement
**de mon corps Dieu ajustait celui d*un rhinoceros,

450 THE SCIENCE OF MECHANICS,

'*en sorte que ses mouvements fussent tellement d'ac-
** cord avec les ordres de mon ame, qu'il levat la patte
'*au moment que je voudrais lever la main, et ainsi

* * des autres operations, ce serait alors mon corps. Je

* * me trouverais subitement dans la forme d*un rhino-
'< ceros au milieu de TAfrique, mais non obstant cela
"mon ime continuerait les m^me operations. J*aurais
"^galement Thonneur d'dcrire k V. A., mais je ne sais
** pas comment elle recevrait mes lettres."

Euier's Ouc would almost imagine that Euler, here, had been

thcoloEic&l

proclivities tempted to play Voltairolt And yet, apposite as was
his criticism in this vital point, the mutual action of
body and soul remained a miracle to him, still. But he
extricates himself, however, from the question of the
freedom of the will, very sophistically. To give some
idea of the kind of questions which a scientist was per-
mitted to treat in those days, it may be remarked that
Euler institutes in his physical ** Letters" investiga-
tions concerning the nature of spirits, the connection
between body and soul, the freedom of the will, the
influence of that freedom on physical occurrences,
prayer, physical and moral evils, the conversion of sin-
ners, and such like topics ; — and this in a treatise full
of clear physical ideas and not devoid of philosophical
ones, where the well-known circle-diagrams of logic
have their birth-place.

Character 3. Let these examples of religious physici^s suffice.

ofihetheo- __ , , , , . • n r ,

logical We have selected them intentionally from among the
the great in- foremost of Scientific discoverers. The theological pro-
clivities which these men followed, belong wholly to
their innermost private life. They tell us openly things
which they are not compelled to tell us, things about
which they might have remained silent. What they
utter are not opinions forced upon them from without ;

FORMAL DEVELOPMENT. 451

they are their own sincere views. They were not con-
scious of any theological constraint. In a court which
harbored a Lamettrie and a Voltaire, Euler had no rea-
son to conceal his real convictions.

According to the modern notion, these men should character

of their age.

at least have seen that the questions they discussed
did not belong under the heads where they put them,
that they were not questions of science. Still, odd as
this contradiction between inherited theological beliefs
and independently created scientific convictions seems
to us, it is no reason for a diminished admiration of
those leaders of scientific thought. Nay, this very fact
is a proof of their stupendous mental power : they were
able, in spite of the contracted horizon of their age, to
which even their own apergus were chiefly limited, to
point out the path to an elevation, where our genera-
tion has attained a freer point of view.

Every unbiassed mind must admit that the age in
which the chief development of the science of mechan-
ics took place, was an £ige of predominantly theological
cast. Theological questions were excited by everything,
and modified everything. No wonder, then, that me-
chanics took the contagion. But the thoroughness with
which theological thought thus permeated scientific
inquiry, will best be seen by an examination of details.

4. The impulse imparted in antiquity to this direc- Galileo's

researches

tion of thought by Hero and Pappus has been alluded on the
to in the preceding chapter. At the beginning of the materials,
seventeenth century we find Galileo occupied with prob-
lems concerning the strength of materials. He shows
that hollow tubes offer a greater resistance to flexure
than solid rods of the same length and the same quantity
of material, and at once applies this discovery to the
explanation of the forms of the bones of animals, which

452 THE SCIENCE OF MECHANICS.

are usually hollow and cylindrical in shape. The phe-
nonnenon is easily illustrated by the comparison of a
flatly folded and a rolled sheet of paper. A horizontal
beam fastened at one extremity and loaded at the other
may be remodelled so as to be thinner at the loaded
end without any loss of stiffness and with a consider-
able saving of material. Galileo determined the form of
a beam of equal resistance at each cross-section. He
also remarked that animals of similar geometrical con-
struction but of considerable difference of size would
comply in very unequal proportions with the laws of
resistance.
Evidences The forms of boues, feathers, stalks, and other or-

of design . 1,1 ....

in nature, ganic Structures, adapted, as they are, m their minut-
est details to the purposes they serve, are highly cal-
culated to make a profound impression on the thinking
beholder, and this fact has again and again been ad-
duced in proof of a supreme wisdom ruling in nature.
Let us examine, for instance, the pinion-feather of a
bird. The quill is a hollow tube diminishing in thick-
ness as we go towards the end, that is, is a body of
equal resistance. Each little blade of the vane re-
peats in miniature the same construction. It would
require considerable technical knowledge even to imi-
tate a feather of this kind, let alone invent it. We
should not forget, however, that scrutiny, or quest of
explanation, not wonder, is the office of science. We
know how Darwin sought to solve these problems, by
the theory of natural selection. That Darwin's solution
is a complete one, may fairly be doubted ; Darwin him-
self questioned it. All external conditions would be
powerless if something were not present that admitted
of variation. But there can be no question that his
theory is the first serious attempt to replace mere won-

FORMAL DEVELOPMENT, 453

der at the adaptations of organic nature by serious in-
quiry into the mode of their origin.

Pappus's ideas concerning the cells of honeycombs The cells r.f

, , . r . , ,. . , , the honey-

were the subject of animated discussion as late as the comb.

eighteenth century. In a treatise, published in 1865,
entitled Homes Without Hands (p. 428), Wood substan-
tially relates the following : " Maraldi had been struck
with the great regularity of the cells of the honey-
comb. He measured the angles of the lozenge-shaped
plates, or rhombs, that form the terminal walls of the
cells, and found them to be respectively 109^28' and
70° 32'. Reaumur, convinced that these angles were in
some way connected with the economy of the cells,
requested the mathematician Konig to calculate the
form of a hexagonal prism terminated by a pyramid
composed of three equal and similar rhombs, which
would give the greatest amount of si>ace with a given
amount of material. The answer was, that the angles
should be 109^26' and 70° 34'. The difference, accord-
ingly, was two minutes. Maclaurin,* dissatisfied with
this agreement, repeated Maraldi's measurements, found
them correct, and discovered, in going over the calcu-
lation, an error in the logarithmic table employed by
Konig. Not the bees, but the mathematicians were
wrong, and the bees had helped to detect the error ! "
Any one who is acquainted with the methods of meas-
uring crystals and has seen the cell of a honeycomb,
with its rough and non -reflective surfaces, will question
whether the measurement of such cells can be executed
with a probable error of only two minutes, f So, we
must take this story as a sort of pious mathematical

* Fkilo*ophic»l TranMoction* for 1743. — Trans.

t But see G. F. Maraldi in the Mimaires d« VacatUmie for 1712. It is, how-
ever, now well known the cells vary considerably. See Chauncey Wright,
PhiUsophieal DiMcuwtions, 1877, p. 3x1. — TVaw^.

454 'J^^F- SCIENCE OF MECHANICS,

fairy-tale, quite apart from the consideration that noth-
ing would follow from it even were it true. Besides,
from a mathematical point of view, the problem is too
imperfectly formulated to enable us to decide the ex-
tent to which the bees have solved it.
Other The ideas of Hero and Fermat, referred to in the

instances.

previous chapter, concerning the motion of light, at
once received from the hands of Leibnitz a theolog-
ical coloring, and played, as has been before mentioned,
a predominant role in the development of the calculus
of variations. In Leibnitz's correspondence with John
Bernoulli, theological questions are repeatedly dis-
cussed in the very midst of mathematical disquisitions.
Their language is not unfrequently couched in biblical
pictures. Leibnitz, for example, says that the problem
of the brachistochrone lured him as the apple had lured
Eve.
Thetheo- Maupertuis, the famous president of the Berlin

nei of the^ Academy, and a friend of Frederick the Great, gave
least ac- a ncw impulse lo the theologising bent of physics by
the enunciation of his principle of least action. In the
treatise which formulated this obscure principle, and
which betrayed in Maupertuis a woeful lack of mathe-
matical accuracy, the author declared his principle to be
the one which best accorded with the wisdom of the
Creator. Maupertuis was an ingenious man, but not a
man of strong, practical sense. This is evidenced by
the schemes he was incessantly devising : his bold prop-
ositions to found a city in which only Latin should be
spoken, to dig a deep hole in the earth to find new
substances, to institute psychological investigations by
means of opium and by the dissection of monkeys, to
explain the formation of the embryo by gravitation, and
so forth. He" was sharply satirised by Voltaire in the

FORMAL DEVELOPMENT. 455

Histoire du docteur Akakia, a work which led, as we
know, to the rupture between Frederick and Voltaire.

Maupertuis's principle would in all probability soon Enier'sre-

- ,r t,-r^f 1 1 tention of

have been forgotten, had Euler not taken up the sug- the thcoioK-

icsil b&sis of

gestion. Euler magnanimously left the principle its this prin-
name, Maupertuis the glory of the invention, and con-
verted it into something new and really serviceable.
What Maupertuis meant to convey is very difficult to
ascertain. What Euler meant may be easily shown by
simple examples. If a body is constrained to move on a
rigid surface, for instance, on the surface of the earth, it
will describe when an impulse is imparted to it, the
shortest path between its initial and' terminal positions.
Any other path that might be prescribed it, would be
longer or would rfequire a greater time. This principle
finds an application in the theory of atmospheric and
oceanic currents. The theological point of view, Euler
retained. He claims it is possible to explain phenomena,
not only from their physical causes^ but also from their
purposes, *'As the construction of the universe is the
''most perfect possible, being the handiwork of an
" all-wise Maker, nothing can be met with in the world
''in which some maximal or minimal property is not
"displayed. There is, consequently, no doubt but
"that all the effects of the world can be derived by
"the method of maxima and minima from their final
"causes as well as from their efficient ones."*

5. Similarly, the notions of the constancy of the
quantity of matter, of the constancy of the quantity of

* ** Qunm enim mundi universi fabrica sit perfectissima, atque a creatore
sapientissimo absoluta, nihil omnino in mundo contingit, in quo non maximi
minimive ratio quaepiam eluceat ; quam ob rem dubium prorsus est nullum,
quin omnes mundi effectus ex causis finalibus, ope method! maximorum et
minimorum, aequo feliciter determinari quaeant, atque ex ipsis causis efficien- *

tibus." {Methodtu invtniendi tineas curuas maximi minimive propitiate
gamdentet, Lausanne, 1744.)

456 THE SCIENCE OF MECIIAXICS.

The central motion, of the indestructibility of work or energy, con-
modern ceptions which completely dominate modern physics,
mainly of all arose under the influence of theological ideas. The

theological . ^- v j ^i • • • • ^^ t

origin. uotious m questiou had their origin in an utterance of
Descartes, before mentioned, in the Principles of Philos-
ophy, agreeably to which the quantity of matter and mo-
tion originally created in the world, — such being the
only course compatible with the constancy of the Crea-
tor,— is always preserved unchanged. The conception
of the manner in which this quantity of motion should
be calculated was very considerably modified in the
progress of the idea from Descartes to Leibnitz, and to
their successors, and as the outcome of these modifi-
cations the doctrine gradually and slowly arose which
is now called the " law of the conservation of energy."
But the theological background of these ideas only
slowly vanished. In fact, at the present day, we still
meet with scientists .who indulge in self-created mys-
ticisms concerning this law.

Gradual During the entire sixteenth and seventeenth centu-

iransition • j i e • i

from the rics, dowu to the close of the eighteenth, the prevail-

theological .....

point of ing inclination of inquirers was, to find in all physical
laws some particular disposition of the Creator. But
a gradual transformation of these views must strike
the attentive observer. Whereas with Descartes and
Leibnitz physics and theology were still greatly inter-
mingled, in the subsequent period a distinct endeavor
is noticeable, not indeed wholly to discard theology,
yet to separate it from purely physical questions. Theo-
logical disquisitions were put at the beginning or rele-
gated to the end of physical treatises. Theological
speculations were restricted, as much as possible, to
the question of creation, that, from this point onward,
the way might be cleared for physics.

view.

FORMAL DEVELOPMENT. 457

Towards the close of the eighteenth century a re- ultimate

. complete

markable change took place, — a change which wasemancipa-
apparentiy an abrupt departure from the current trend physics

from tlieol*

of thought, but in reality was the logical outcome of ogy.
the development indicated. After an attemi>t in a
youthful work to found mechanics on Euler' s principle
of least action, Lagrange, in a subsequent treatment
of the subject, declared his intention of utterly disre-
garding theological and metaphysical speculations, as
in their nature precarious and foreign to science. He
erected a new mechanical system on entirely different
foundations, and no one conversant with the subject
will dispute its excellencies. All subsequent scientists
of eminence accepted Lagrange's View, and the pres-
ent attitude of physics to theology was thus substan-
tially determined.

6. The idea that theology and physics are two dis- The mod-
tinct branches of knowledge, thus took, from its first aiwayj «*»«
germination in Copernicus till its. final proniulgation !*»««.'«**««
by Lagrange, almost two centuries to attain clearness
in the minds of investigators. At the same time it
cannot be denied that this truth was always clear to
the greatest minds, like Newton. Newton never, de-
spite his profound religiosity, mingled theology with
the questions of science. True, even he concludes his
Optics, whilst on its last pages his clear and luminous
intellect still shines, with an exclamation of humble
contrition at the vanity of all earthly things. But his
optical researches proper, in contrast to those of Leib-
nitz, contain not a trace of theology. The same may
be said of Galileo and Huygens. Their writings con-
form almost absolutely to the point of view of La-
grange, and may be accepted in this respect as class-
ical. But the general views and tendencies of an age

458 THE SCIENCE OF MECHANICS.

must not be judged by its greatest, but by its average,
minds.
The theo- To Comprehend the process here portrayed, the gen-

lOf^lCSl COQ-

ceptionof eral condition of affairs in these times must be consid*

the world

natural and ered. It stauds to reasou that in a stage of civilisation
able. in which religion is almost the sole education, and the

only theory of the world, people would naturally look
at things in a theological point of view, and that they
would believe that this view was possessed of compe-
tency in all fields of research. If we transport ourselves
back to the time when people played the organ with
their fists, when they had to have the multiplication table
visibly before them to calculate, when they did so much
with their hands that people now-a-days do with their
heads, we shall not demand of such a time that it
should critically put to the test its own views and the-
ories. With the widening of the intellectual horizon
through the great geographical, technical, and scien-
tific discoveries and inventions of the fifteenth and six-
teenth centuries, with the opening up of provinces in
which it was impossible to make any progress with the
old conception of things, simply because it had been
formed prior to the knowledge of these provinces, this
bias of the mind gradually and slowly vanished. The
great freedom of thought which appears in isolated
cases in the early middle ages, first in poets and then
in scientists, will always be hard to understand. The en-
lightenment of those days must have been the work of a
few very extraordinary minds, and can have been bound
to the views of the people at large by but very slender
threads, more fitted to disturb those views than to re-
form them. Rationalism does not seem to have gained
a broad theatre of action till the literature of the eigh-
teenth century. Humanistic, philosophical, historical,

FORMAL DEVELOPMENT. 459

and physical science here met and gave each other
mutual encouragement. All who have experienced, in
part, in its literature, this wonderful emancipation of
the human intellect, will feel during their whole lives a
deep, elegiacal regret for the eighteenth century.

7. The old point of view, then, is abandoned. Its The en-
history is now detectible only in the form of the me- mem of the
chanical principles. And this form will remain strange
to us as long as we neglect its origin. The theological
conception of things gradually gave way to a more
rigid conception ; and this was accompanied with a
considerable gain in enlightenment, as we shall now
briefly indicate.

When we say light travels by the paths of shortest
time, we grasp by such an expression many things.
But we do not know as yet why light prefers paths of
shortest time. We forego all further knowledge of the
phenomenon, if we find the reason in the Creator's wis-
dom. We of to-day know, that light travels by all
paths, but that only on the paths of shortest time do
the waves of light so intensify each other that a per-
ceptible result is produced. Light, accordingly, only
appears to travel by the paths of shortest time. After Extrava-
the prejudice which prevailed on these questions had weii as

• J- ^ 1 J- J • economy in

been removed, cases were immediately discovered in nature,
which by the side of the supposed economy of nature
the most striking extravagance was displayed. Cases
of this kind have, for example, been pointed out by
Jacobi in connection with Euler's principle of least ac-
tion. A great many natural phenomena accordingly
produce the impression of economy, simply because
they visibly appear only when by accident an econom-
ical accumulation of effects take place. This is the
same idea in the province of inorganic nature that Dar-

46o THE SCIENCE OF MECHANICS.

win worked out in the domain of organic nature. We
facilitate instinctively our comprehension of nature by
applying to it the economical ideas with which we are
familiar.
Expiana- Often the phenomena of nature exhibit maximal

tionofmax- ... . , , ,

imai and or mmimal properties because when these greatest or
effects. least properties have been established the causes of all
further alteration are removed. The catenary gives
the lowest point of the centre of gravity for the simple
reason that when that point has been reached all fur-
ther descent of the system's parts is impossible. Li-
quids exclusively subjected to the action of molecular
forces exhibit a minimum of superficial area, because
stable equilibrium can only subsist when the molecular
forces are able to effect no further diminution of super-
ficial area. The important thing, therefctre, is not the
maximum or minimum, but the removal of work ; work
being the factor determinative of the alteration. It
sounds much less imposing but is much more elucida-
tory, much more correct and comprehensive, instead
of speaking of the economical tendencies of nature, to
say : "So much and so much only occurs as in virtue
of the forces and circumstances involved can occur."
Points of The question may now justly be asked, If the point

the theoiog- of view of thcology which led to the enunciation of the
scientific principles of mechanics was utterly wrong, how comes
tions. it that the principles themselves are in all substantial
points correct ? The answer is easy. In the first place,
the theological view did not supply the contents of the
principles, but simply determined their guise\ their mat-
ter was derived from experience. A similar influence
would have been exercised by any other dominant type
of thought, by a commercial attitude, for instance, such
as presumably had its effect on Stevinus's thinking. In

FORMAL DEVELOPMENT. 461

the second place, the theological conception of nature
itself owes its origin to an endeavor to obtain a more
comprehensive view of the world ; — the very same en-
deavor that is at the bottom of physical science. Hence,
even admitting that the physical philosophy of theology
is a fruitless achievement, a reversion to a lower state of
scientific culture, we still need not repudiate the sound
root from which it has sprung and which is not differ-
ent from that of true physical inquiry.

In fact, science can accomplish nothing by the con- Necessity

of a con-
sideration of individual facts ; from time to time it must stant con-

cast Its glance at the world as a whole, Galileo s of the ah,

. . in research

laws of falling bodies, Huygens's principle of vis viva,
the principle of virtual velocities, nay, even the con-
cept of mass, could not, as we saw, be obtained, ex-
cept by the alternate consideration of individual facts
and of nature as a totality. We may, in our men-
tal reconstruction of mechanical processes, start from
the properties of isolated masses (from the elementary
or differential laws), and so compose our pictures of
the processes ; or, we may hold fast to the properties
of the system as a whole (abide by the integral laws).
Since, however, the properties of one mass always in-
clude relations to other masses, (for instance, in ve-
locity and acceleration a relation of time is involved,
that is, a connection with the whole world,) it is mani-
fest that purely differential, or elementary, laws do not
exist. It would be illogical, accordingly, to exclude
as less certain this necessary view of the All, or of the
more general properties of nature, from our studies.
The more general a new principle is and the wider its
scope, the more perfect tests will, in view of the possi-
bility of error, be demanded of it.

The conception of a will and intelligence active in

462 THE SCIENCE OF MECHANICS,

Pasan ideas nature IS by no means the exclusive property of Chris-
tices rife in tian monotheism. On the contrary, this idea is a quite

the modem , ... . i r • i • -n*

world. lamihar one to paganism and fetishism. Paganism,
however, finds this will and intelligence entirely in in-
dividual phenomena, while monotheism seeks it in the
All. Moreover, a pure monotheism does not exist.
The Jewish monotheism of the Bible is by no means
free from belief in demons, sorcerers, and witches ;
and the Christian monotheism of mediaeval times is
even richer in these pagan conceptions. We shall not
speak of the brutal amusement in which church and
state indulged in the torture and burning of witches,
and which was undoubtedly provoked, in the majority
of cases, not by avarice but by the prevalence of the
ideas mentioned. In his instructive work on Primitive
Culture Tylor has studied the sorcery, superstitions,
and miracle- belief of savage peoples, and compared
them with the opinions current in mediaeval times con-
cerning witchcraft. The similarity is indeed striking.
The burning of witches, which was so frequent in
Europe in the sixteenth and seventeenth centuries, is
to-day vigorously conducted in Central Africa. Even
now and in civilised countries and among cultivated
people traces of these conditions, as Tylor shows, still
exist in a multitude of usages, the sense of which, with
our altered point of view, has been forever lost.

8. Physical science rid itself only very slowly of
these conceptions. The celebrated work of Giambatista
della Porta, Magia naturalis^ which appeared in 1558,
though it announces important physical discoveries, is
yet filled with stuff about magic practices and demono-
logical arts of all kinds little better than those of a red-
skin medicine-man. Not till the appearance of Gil-
bert's work. De magnete (in r6oo), was any kind of re-

FORMAL DEVELOPMENT, 463

striction. placed on this tendency of thought. When we Animistic

_ . . notions in

reflect that even Luther is said to have had personal science,
encounters with the Devil, that Kepler, whose aunt had
been burned as a witch and whose mother came near
meeting the same fate, said that witchcraft could not
be denied, and dreaded to express his real opinion of
astrology, we can vividly picture tp ourselves the
thought of less enlightened minds of those ages.

Modern physical science also shows traces of fetish-
ism, as Tylor >vell remarks, in its "forces." And the
hobgoblin practices of modern spiritualism are ample
evidence that the conceptions of paganism have not
been overcome even by the cultured society of to-day.

It is natural that these ideas so obstinately assert
themselves. Of the many impulses that rule man
with demoniacal power, that nourish, preserve, and
propagate him, without his knowledge or supervision,
of these impulses of which the middle ages present
such great pathological excesses, only the smallest
part is accessible to scientific analysis and conceptual
knowledge. The fundamental character of all these
instincts is the feeling of our oneness and sameness
with nature ; a feeling that at times can be silenced
but never eradicated by absorbing intellectual occupa-
tions, and which certainly has a sound basis, no matter
to what religious absurdities it may have given rise.

9. The French encyclopaedists of the eighteenth
century imagined they were not far from a final ex-
planation of the world by physical and mechanical prin-
ciples ; Laplace even conceived a mind competent to
foretell the progress of nature for all eternity, if but the
masses, their positions, and initial velocities were given.
In the eighteenth century, this joyful overestimation of
the scope of the new physico-mechanical ideas is par-

464

THE SCIENCE OF MECHANICS,

Overcsti-
mation of
the me-
chanical
view.

Pretensions
and atti-
tude of
physical
science.

donabie. Indeed, it is a refreshing, noble, and ele-
vating spectacle ; and we can deeply sympathise with
this expression of intellectual joy, so unique in history.
But now, after a century has elapsed, after our judg-
ment has grown more sober, the world-conception of the
encyclopaedists appears to us as a viechanical mythology
in contrast to the animistic of the old religions. Both
views contain undue and fantastical exaggerations of
an incomplete perception. Careful physical research
will lead, however, to an analysis of our sensations.
We shall then discover that our hunger is not so essen-
tially different from the tendency of sulphuric acid for
zinc, and our will not so greatly different from the
pressure of a stone, as now appears. We shall again
feel ourselves nearer nature, without its being neces-
sary that we should resolve ourselves into a nebulous
and mystical mass of molecules, or make nature a
haunt of hobgoblins. The direction in which this en-
lightenment is to be looked for, as the result of long
and painstaking research, can of course only be sur-
mised. To anticipate the result, or even to attempt to
introduce it into any scientific investigation of to-day,
would be mythology, not science.

Physical science does not pretend to be a complete
view of the world ; it simply claims that it is working
toward such a complete view in the future. The high-
est ^ilosophy of the scientific investigator is precisely
this toleration of an incomplete conception of the world
and the preference for it rather than an apparently per-
fect, but inadequate conception. Our religious opin-
ions are always our own private affair, as long as we do
not obtrude them upon others and do not apply them
to things which come under the jurisdiction of a differ-
ent tribunal. Physical inquirers themselves entertain

FORMAL DEVELOPMENT. 465

the most diverse opinions on this subject, according to
the range of their intellects and their estimation of the
consequences.

Physical science makes no investigation at all into
things that are absolutely inaccessible to exact investi-
gation, or as yet inaccessible to it. But should prov-
inces ever be thrown open to exact research which are
now closed to it, no well-organised man, no one who
cherishes honest intentions towards himself and others,
will any longer then hesitate to countenance inquiry
with a view to exchanging his opinion regarding such
provinces for positive knowledge of them.

When, to-day, we see society waver, see it change Results of

. . . ^. ,. . . theinconi-

its Views on the same question according to its mood and pietenessof

OQr view of

the events of the week, like the register of an organ, when the world.
we behold the profound mental anguish which is thus
produced, we should know that this is the natural and
necessary outcome of the incompleteness and transi-
tional character of our philosophy. A competent view
of the world can never be got as a gift ; we must ac-
quire it by hard work. And only by granting free sway
to reason and experience in the provinces in which they
alone are determinative, shall we, to the weal of man-
kind, approach, slowly, gradually, but surely, to that
ideal of a monistic view of the world which is alone
compatible with the economy of a sound mind.

III.

ANALYTICAL MECHANICS.

I. The mechanics of Newton are purely geometrical. The peo-

mctncsil

He deduces his theorems from his initial assumptions mechanics

, . - . - . __. of Newton.

entirely by means of geometrical constructions. His
procedure is frequently so artificial that, as Laplace

466

THE SCIENCE OF MECHANICS.

Analytic
mechanics.

Euler and
Maclau-
rin's con-
tributions.

Lagrange's
perfection
of the
science.

remarked, it is unlikely that the propositions were dis-
covered in that way. We notice, moreover, that the
expositions of Newton are not as candid as those of
Galileo and Huygens. Newton's is the so-called syn-
thetic method of the ancient geometers.

When we deduce results from given suppositions,
the procedure is called synthetic. When we seek the
conditions of a proposition or of the properties of a fig-
ure, the procedure is analytic. The practice of the latter
method became usual largely in consequence of the
application of algebra to geometry. It has become
customary, therefore, to call the algebraical method
generally, the analytical. The term ''analytical me-
chanics," which is contrasted with the synthetical, or
geometrical, mechanics of Newton, is the exact equiva-
lent of the phrase "algebraical mechanics."

2. The foundations of analytical mechanics were
laid by Euler {Mechanica, sive Motus Scientia Analytice
Exposita, St. Petersburg, 1736). But while Euler's
method, in its resolution of curvilinear forces into tan-
gential and normal components, still bears a trace of
the old geometrical modes, the procedure of Maclaurin
{A Complete System of Fluxions ^ Edinburgh, 1742) marks
a very important advance. This author resolves all
forces in three fixed directions, and thus invests the
computations of this subject with a high degree of
symmetry and perspicuity.

3. Analytical mechanics, however, was brought to
its highest degree of perfection by Lagrange. La-
grange's aim is {Mecanique analytique^ Paris, 1788) to
dispose once for all oi the reasoning necessary to resolve
mechanical problems, by embodying as much as pos-
sible of it in a single formula. This he did. Every case
that presents itself can now be dealt with by a very

FORMAL DEVELOPMENT, 467

simple, highly symmetrical and perspicuous schema ;
and whatever reasoning is left is performed by purely
mechanical methods. The mechanics of Lagrange
is a stupendous contribution to the economy of
thought.

In statics, Lagrange starts from the principle ofstaUcs

.... ,^ , , ... founded on

Virtual velocities. On a number of material points the princi-
ples of vir-

Wj, Wg, ///g. . . ., definitely connected with one another, tuai veioci-
are impressed the forces P^, P^y P^, . , , If these
points receive any infinitely small displacements /j,
/2> /s • • • • compatible with the connections of the sys-
tem, then for equilibrium JSPp = 0 ; where the well-
known exception in which the equality passes into an
inequality is left out of account.

Now refer the whole system to a set of rectangular
codrdinates. Let the coordinates of the material points
be jCj, ^j, 2j, jTg, ^2> ^2 • • • • Resolve the forces into
the components X^, V^, Z^, X^, V^, Z^, . . . parallel
to the axes of coordinates, and the displacements into
the displacements dx^^, dy^y dz^, 6x2, dy^s ^^2' • • •>
also parallel to the axes. In the determination of the
work done only the displacements of the point of appli-
cation in the direction of each force-component need
be considered for that component, and the expression
of the principle accordingly is

2(XSx+ V6y + Zdz) = 0 (1)

where the appropriate indices are to be inserted for
the points, and the final expressions summed.

The fundamental formula of dynamics is derived Dynamics

on the prin-

from D'Alembert's principle. On the material points cipie of

D'Aleni'

m^, m^, W3 . . . ., having the codrdinates x^^ y^, z^, x^, bert.
^2, 2^2 ... . the force-components AT^, K^, Z^, A'g, 1^2,
Zj . . . . act. But, owing to the connections of the

468 THE SCIENCE OF MECHANICS,

system's parts, the masses undergo accelerations, which
are those of the forces.

d^x^ d^y^ d^z^

These are called the effective forces. But the impressed
forces, that is, the forces which exist by virtue of the
laws of physics, X, Y, Z. , , . and the negative of these
effective forces are, owing to the connections of the
system, in equilibrium. Applying, accordingly, the
principle of virtual velocities, we get

Discussion 4. Thus, Lagrange conforms to tradition in making

of La-

grange's statics precede dynamics. He was by no means com-
pelled to do so. On the contrary, he might, with equal
propriety, have started from the proposition that the
connections, neglecting their straining, perform no
work, or that all the possible work of the system is due
to the impressed forces. In the latter case he would
have begun with equation (2), which expresses this
fact, and which, for equilibrium (or non-accelerated
motion) reduces itself to (i) as a particular case. This
would have made analytical mechanics, as a system,
even more logical.

Equation (i), which for the case of equilibrium
makes the element of the work corresponding to the
assumed displacement = 0, gives readily the results
discussed in page 69. If

dV dV dV

^~ dx' dy' dz'

FORMAL DEVELOPMENT, 469

that is to say, if X, F, Z are the partial differential co-
efficients of one and the same function of the coordi-
nates of position, the whole expression under the sign
of summation is the total variation, d V, of F, If the
latter is = 0, Kis in general a maximum or a minimum.

5. We will now illustrate the use of equation (i) by indication

. . . . ofthegen-

a simple example. If all the points of application of the erai steps

.1 r 1 1 • 'orthesolu-

forces are mdependent of each other, no problem is tion of stat-

. . . ... *cal prob-

presented. Each point is then in equilibrium onlyiems.
when the forces impressed on it, and consequently
their components, are == 0. All the displacements dx,
6}'y Sz, . , , are then wholly arbitrary, and equation
(i) can subsist only provided the coefficients of all the
displacements dx, 6y, 6z, , . . are equal to zero.

But if equations obtain between the coordinates of
the several points, that is to say, if the points are sub-
ject to mutual constraints, the equations so obtaining
will be of the form F(x^y y^y z^, x^, y^, z^. . . .) = 0,
or, more briefly, of the form jF=0, Then equations
also obtain between the displacements, of the form

^^; ^^^ + <^ ^-^'^ + ^ ^^^ + ^:c/^« + •••• = ^'

which we shall briefly designate asZ>/^=0. If the
system consist of n points, we shall have 3 n coordi-
nates, and equation (i) will contain ^n magnitudes
dxy Sy, dz, , . , If, further, between the coordinates
fn equations of the form jF=0 subsist, then m equa-
tions of the form DF= 0 will be simultaneously given
between the variations Sx, 6y, 6z . , , . By these
equations m variations can be expressed in terms of the
remainder, and so inserted in equation (i). In (i),
therefore, there are left 3« — m arbitrary displace-
ments, whose coefficients are put = 0. There are thus

470

THE SCIENCE OF MECHANICS.

A statical
example.

obtained between the forces and the co6rdinates 3 » — m
equations, to which the m equations (-^= 0) must be
added. We have, accordingly, in all, 3« equations,
which are sufficient to determine the. 3;! coordinates of
the position of equilibrium, provided the forces are
given and only they^rw of the system's equilibrium is
sought.

But if the form of the system is given and the forces
are sought that maintain equilibrium, the question is
indeterminate. We have then, to determine 3 n force-
components, only 3» — m equa-
tions ; the m equations (J^ = 0)
not containing the force-compo-
nents.

As an example of this man-
ner of treatment we shall select
a lever OM^za^ free to rotate
about the origin of co6rdinates
in the plane XYy and having at its end a second, simi-
lar lever MN^=^b, At M and Ny the coordinates of
which we shall call x^ y and x^^ y\y the forces X^ Kand
X^y Fj are applied. Equation (i), then, has the form

X6x^X^6x^ + Ydy^ Y^dy^=^ • • • (3)

Of the form I*'=0 two equations here exist ; namely,

^x,-xy + (y^-yy- fi2^o f ^ ^

The equations I>F= 0, accordingly, are

xSx -\- ySy = 0 J

(^1 — X) 6x^ — {x^ — x)dx-\r d —y^ ^y^—Y • (5)

{y^ —y) ^y = ^ )

Here, two of the variations in (5) can be expressed
in terms of the others and introduced in (3). Also for

Pig. 232.

FORMAL DEVELOPMENT. 471

purposes of elimination Lagrange employed a per- Lagrangc/a
fectly uniform and systematic procedure, which maynatecocffi-
be pursued quite mechanically, without reflection. We
shall use it here. It consists in multiplying each of .the
equations (5) by an indeterminate coefficient A, /i, and
adding each in this form to (3). So doing, we obtain

\Y^\y-lx^^-y)-\dy ^ly,-^ pi^y -y)-\dy, f-"*

The coefficients of the four displacements may now
be put (Jirectly = 0. For two displacements are ar-
bitrary, and the two remaining coefficients may be
made equal to zero by the appropriate choice of A and
}x — which is tantamount to an elimination of the two
remaining displacements.

We have, therefore, the four equations

X -\- \x — A'G'^'i — jf) = 0

-^1 +)"(^i — •^) = ^
Y\\y-pL^y^-y') = ^

We shall first assume that the coordinates are given,
and seek the forces that maintain equilibrium. The
values of \ and pi are each determined by equating to
zero two coefficients. We get from the second and
fourth equations,

p, = i-, and /i = ^ ,

x,—x y^—y

whence

y,~y,-y ^ ^

that is to say, the total component force impressed at
N has the direction MN. From the first and third
equations we get

(6)

47a THE SCIENCE OF MECHANICS.

Their en.. A = --^-i-^^tZlfl, A = --^+-^-^-*''--::^,
ployment in x Y

the deter-

{Jj.°J{j°j° °' and from these by simple reduction

equation. X+X, X

Y+y;=y ^^^

that is to say, the resultant of the forces applied at Af
and N acts in the direction OM, *

The four force-components are accordingly subject
to only two conditions, (7) and (8). The problem, con-
sequently, is an indeterminate one ; as it must be from
the nature of the case ; for equilibrium does not depend
upon the absolute magnitudes of the forces, but upon
their directions and relations.

If we assume that the forces are given and seek the
four coordinates, we treat equations (6) in exactly the
same manner. Only, we can now make use, in addi-
tion, of equations (4). Accordingly, we have, upon the
elimination of X and fji, equations (7) and (8) and two
equations (4). From these the following, which fully
solve the problem, are readily deduced •

y =

V{X+X^)^-\-{Y^YJ

* The mechanical interpretation of the indeterminate coefficients At j^ may
be shown as follows. Equations (6) express the equilibrium of Xvio/ree points
on which in addition to X, V, X^, V'^ other forces act which answer to the re-
maining expressions and just destroy A", l', Jfj, K^. The point N^ for example,
is in equilibrium if Xi is destroyed by a force fi [x^ — x), undetermined as yet
in magnitude, and V\ by a force H [y\ — y). This supplementary force is due
to the constraints. Its direction is determined ; though its magnitude is not.
If we call the angle which it makes with the axis of abscissas a, we shall have

tana = '';^''"^l^^>->'-

that is to say. the force due to the connections acts in the direction of i.

FORMAL DEVELOPMENT, 473

X^ = ^ — ^^ 1 ^ Character

lein.

Simple as this example is, it is yet sufficient to give
us a distinct idea of the character and significance of
Lagrange's method. The mechanism of this method is
excogitated once for all, and in its application to par-
ticular cases scarcely any additional thinking is re-
quired. The simplicity of the example here selected
being such that it can be solved by a mere glance at
the figure, we have, in our study of the method, the
advantage of a ready verification at every step.

6. We will now illustrate the application of equa- General
tion (2), which is Lagrange's form of statement of theTsoiution
D*Alembert*s principle. There is no problem whenfcai^ro™

IfilXlS

the masses move quite independently of one another.
Each mass yields to the forces applied to it ; the va-
riations dXf dy, dz . , , , are wholly arbitrary, and each
coefficient may be singly put == 0. For the motion of
n masses we thus obtain 3 n simul-
taneous differential equations.

But if equations of condition
(/^= 0) obtain between the coordi-
nates, these equations will lead to
others {DF--^^) between the dis-
placements or variations. With the

. F»R. 233-

latter we proceed exactly as in the
application of equation (i). Only it must be noted
here that the equations 1^=0 must eventually be em-
ployed in their undifferentiated as well as in their dif-
ferentiated form, as will best be seen from the follow-
ing example.

474 THE SCIENCE OF MECHANICS,

A dynam- A hcavy material point «r, lying in a vertical plane

pie. XY, is free to move on a straight line, y =i axj inclined

at an angle to the horizon. (Fig. 233.) Here equa-
tion (2) becomes

and, since A!'= 0, and Y= — mg^ also

'J/l'^* + (^+S>^'=« (^)

The place of -^ = 0 is taken by

y = ax (10)

and for DF == 0 we have

dy = a6x.

Equation (9), accordingly, since 6y drops out and
djc is arbitrary, passes into the form

^x f d^}\

By the differentiation of (10), or {F= 0), we have

d^y d^x I

and, consequently,

d'^x . ( . d^x\

Then, by the inte.gration of (11), we obtain

— a /2

and

where b and c are constants Qf integration, determined
by the initial position and velbcity of m. This result
can also be easily found by the direct method.

FORMAL DEVELOPMENT, 475

Some care is necessary in the application of equa- a modifica-

tion (i) \i F'=i\S contains the time. The procedure m example,
such cases may be illustrated by the following example.
Imagine in the preceding case the straight line on
which m descends to move vertically upwards with the
acceleration y* We start again from equation (9)

d'^x ^ , / . d^y\ ^

/'= 0 is here replaced by

y = ax^Y% •■■■'■ (12)

To form DF= 0, we vary (12) only with respect to x
and y, for we are concerned here only with the possible
displacement of the system in its position at any given
instant^ and not with the displacement that actually
takes place in time. We put, therefore, as in the pre-
vious case,

dy =^ a dx,

and obtain, as before,

d^x I . d^y\ ^ ,,„,

But to get an equation in x alone, we have, since x
and >> are connected in (13) by the actual motion, to
differentiate (12) with respect to / and employ the re-
sulting equation

///2 ■" ^ dt^~ "*" ^
for substitution in (13). In this way the equation

is obtained, which, integrated, gives

476

THE SCIENCE OF MECHANICS,

y =

y—

u + x)

-|- <z^/ -j- ac.

If a weightless body w lie on the moving straight
line, we obtain these equations

■^ = rT^/2 + ^^' + ^^'

— results which are readily understood, when we re-
flect that, on a straight line moving upwards with the
acceleration ^, m behaves as if it were affected with a
downward acceleration y on the straight line at rest.
Discussion 7- The procedure with equation ( 1 2) in the preced-
ffied exam- ing example may be rendered somewhat clearer by the
^®" following consideration. Equation (2), D*Alembert's

principle, asserts, that all the work
that can be done in the displacement
of a system is done by the impressed
forces and not by the connections. This
is evident, since the rigidity of the con-
nections allows no changes in the rela-
tive positions which would be neces-
sary for any alteration in the potentials of the elastic
forces. But this ceases to be true when the connec-
tions undergo changes in time. In this case, the changes
of the connections perform work, and we can then ap-
ply equation (2) to the displacements that actually take
place only provided we add to the impressed forces the
forces that produce the changes of the connections.

A heavy mass in is free to move on a straight line
parallel to C?F(Fig. 234.) Let this line be subject to

Fig. *34-

FORMAL DEVELOPMENT. 477

a forced acceleration in the direction of x, such that illustration

of the mod-

the equation jF=0 becomes ifiedexam-

^ pie.

^ = y-l> 0-^)

D'Alembert*s principle again gives equation (9).
But, since from Z>/^= 0 it follows here that dx = 0,
this equation reduces itself to

^+S')*^=' ^''^

in which 6y is wholly arbitrary. Wherefore,

and

to which must be supplied (14) or

It is patent that (15) does not assign the total work
of the displacement that actually takes place, but only
that of some possible displacement on the straight line
conceived, for the moment, as fixed.

If we imagine the straight line massless, and cause
it to travel parallel to itself in some guiding mechan-
ism moved by a force niyy equation (2) will be re-
placed by

my — m ~^^^^ 6x + [— mg — m -^^j^y = 0,

and since dx^ dy are wholly arbitrary here, we obtain

the two equations

inx .
V ^0

478 THE SCIENCE OF MECHANICS.

which give the same results as before. The apparently
different mode of treatment of these cases is simply the
result of a slight inconsistency, springing from the fact
that all the forces involved are, for reasons facilitating
calculation, not included in the consideration at the
outset, but a portion is left to be dealt with subse-
quently.
Deduction 8. As the different mechanical principles only ex-
cipie^o?"/* press different aspects of the same fact, any one of
Lagrange's them is easily deducible from any other ; as we shall
tardynam- HOW illustrate by developing the principle of %ns inva
ica^equa- £j.qj^ equation (2) of page 468. Equation (2) refers to
instantaneously possible displacements, that is, to "vir-
tual " displacements. But when the connections of a
system are independent of the time, the motions that
actually take place 2X^ "virtual" displacements. Conse-
quently the principle may be applied to actual motions.
For dxy 6y, dz, we may, accordingly, write dx, dy,
dzy the displacements which take place in time, and
put

^ iXdx + Ydy + Zdz^ =

^ (d^x . ,d^y. , d^z .
:2fn\^--.dx^rj,,dy^^^^dz

' The expression to the right may, by introducing for
dx^ {dxjdf) dt and so forth, and by denoting the velo-
city by ?;, also be written

^ Id'^ x dx .^ , d^y dy . , d'^ z dz , \
\dt^ dt ^ dt^ dt ~ dt^ dt

^d2m

a AM 2

FORMAL DEVELOPMENT. 479

Also in the expression to the left, {dxjdf) dt may be Force-
written for dx. But this gives

J^ {Xdx + Ydy + Zdz) = 2im (r^ _ ^.2)^

where Vq denotes the velocity at the beginning and 7^
the velocity at the end of the motion. The integral to the
left can always be found if we can reduce it to a single
variable, that is to say, if we know the course of the
motion in time or the paths which the movable points
describe. If, however, X, Y, Z are the partial differ-
ential coefficients of the same function 6^of coordinates,
if, that is to say,

dU dU dU

^^ dx' 77' 7F'

as is always the case when only central forces are in-
volved, this reduction is unnecessary. The entire ex-
pression to the left is then a complete differential. And
we have

which is to say, the difference of the force-functions
(or work) at the beginning and the end of the motion
is equal to the difference of the vires vivce at the be-
ginning and the end of the motion. The vires vivce are
in such case also functions of the coordinates.

In the case of a body movable in the plane of X
and Ksuppose, for example, X=^ — j, K= — x\ we
then have

J^'—ydx — xdy) = —Jd^xy) =

^oyo — ^y= \fn{y^ — vX),

But if A" = — fl, K= — jc, the integral to the left is
— Ha dx -{■ ^ ^y)' This integral can be assigned the
moment we know the path the body has traversed, that

480

THE SCIENCE OF MECHANICS.

Essential
character
of analyt-
ical me-
chanics.

is, if y is determined a function of x. If, for example,
y=px^y the integral would become

-J {a + 2px^) dx ^. a (.r, — x) ^ ^/^ (^o^-_^)_^

The difference of these two cases is, that in the first
the work is simply a function of coordinates, that a
force-function exists, that the element of the work is a
complete differential, and the work consequently is dor
termined by the initial and final values of the coordi-
nates, while in the second case it is dependent on the
entire path described.

9. These simple examples, in themselves present-
ing no difficulties, will doubtless suffice to illustrate the
general nature of the operations of analytical mechan-
ics. No fundamental light can be expected from this
branch of mechanics. On the contrary, the discovery
of matters of principle must be substantially completed
before we can think of framing analytical mechanics ;
the sole aim of which is a perfect practical mastery of
problems. Whosoever mistakes this situation, will
never comprehend Lagrange's great performance, which
here too is essentially of an economical ch^x^iCter. Poin-
sot did not altogether escape this error.

It remains to be mentioned that as the result of the
labors of Mobius, Hamilton, Grassmann, and others, a
new transformation of mechanics is preparing. These
inquirers have developed mathematical conceptions
that conform more exactly and directly to our geomet-
rical ideas than do the conceptions of common analyt-
ical geometry ; and the advantages of analytical gene-
rality and direct geometrical insight are thus united.
But this transformation, of course, lies, as yet, beyond
the limits of an historical exposition.

FORMAL DEVELOPMENT, 481

IV.
THE ECONOMY OF SCIENCE.

I. It is the object of science to replace, or save^ ex- The basis

,, J. , ..'. , f of science,

perienceSy by the reproduction and anticipation of facts economy of
in thought. Memory is handier than experience, and
often answers the same purpose. This economical
office of science, which fills its whole life, is apparent
at first glance ; and with its full recognition all mys-
ticism in science disappears.

Science is communicated by instruction, in order
that one man may profit by the experience of another
and be spared the trouble of accumulating it for him-
self ; and thus, to spare posterity, the experiences of
whole generations are stored up in libraries.

Language, the instrument of this communication, The eco-

...tr *i ^* T> * nomical

IS itself an economical contrivance. Experiences are character
analysed, or broken up, into simpler and more familiar ^agei
experiences, and then symbolised at some sacrifice of
precision. The symbols of speech are as yet restricted
in their use within national boundaries, and doubtless
will long remain so. But written language is gradually
being metamorphosed into an ideal universal character.
It is certainly no longer a mere transcript of speech.
Numerals, algebraic signs, chemical symbols, musical
notes, phonetic alphabets, may be regarded as parts
already formed of this universal character of the fu-
ture ; they are, to some extent, decidedly conceptual,
and of almost general international use. The analysis
of colors, physical and physiological, is already far
enough advanced to render an international system of
color-signs perfectly practical. In Chinese writing,

482 THE SCIENCE OF MECHANICS.

Possibility we have an actual example of a true ideographic Ian-

of a univer- j i • i • i • rr

sal Ian- guage, prouounced diversely m different provinces, yet
everywhere carrying the same meaning. Were the
system and its signs only of a simpler character, the
use of Chinese writing might become universal. The
dropping of unmeaning and needless accidents of gram-
mar, as English mostly drops them, would be quite
requisite to the adoption of such a system. But uni-
versality would not be the sole merit of such a char-
acter ; since to read it would be to understand it. Our
children often read what they do not understand ; but
that which a Chinaman cannot understand, he is pre-
cluded from reading.
Econom- 2. In the reproduction of facts in thought, we

tero*fa\"*^ never reproduce the facts in full, but only that side of
senialfons them which is important to us, moved to this directly
world. or indirectly by a practical interest. Our reproductions
are invariably abstractions. Here again is an econom-
ical tendency.

Nature is composed of sensations as its elements.
Primitive man, however, first picks out certain com-
pounds of these elements — those namely that are re-
latively permanent and of greater importance to him.
The first and oldest words are names of ** things."
Even here, there is an abstractive process, an abstrac-
tion from the surroundings of the things, and from the
continual small changes which these compound sensa-
tions undergo, which being practically unimportant are
not noticed. No inalterable thing exists. The thing
is an abstraction, the name a symbol, for a compound
of elements from whose changes we abstract. The
reasoR we assign a single word to a whole compound is
that we need to suggest all the constituent sensations
at once. When, later, we come to remark the change-

FORMAL DEVELOPMENT. 483

ableness, we cannot at the same time hold fast to the
idea of the thing's permanence, unless we have recourse
to the conception of a thing-in-itself, or other such like
absurdity. Sensations are not signs of things ; but, on
the contrary, a thing is a thought-symbol for a com-
pound sensation of relative fixedness. Properly speak-
ing the world is not composed of ** things" as its ele-
ments, but of colors, tones, pressures, spaces, times,
in short what we ordinarily call individual sensations.

The whole operation is a mere affair of economy.
In the reproduction of facts, we begin with the more
durable and familiar compounds, and supplement these
later with the unusual by way of corrections. Thus,
we speak of a perforated cylinder, of a cube with bev-
eled edges, expressions involving contradictions, un-
less we accept the view here taken. All judgments are
such amplifications and corrections of ideas already
admitted.

3. In speaking of cause and effect we arbitrarily The ideas
give relief to those elements to whose connection we effect,
have to attend in the reproduction of a fact in the re-
spect in which it is important to us. There is no cause
nor effect in nature ; nature has but an individual exis-
tence ; nature simply is. Recurrences of like cases in
which A is always connected with B^ that is, like results
under like circumstances, that is again, the essence of the
connection of cause and effect, exist but in the abstrac-
tion which we perform for the purpose of mentally re-
producing the facts. Let a fact become familiar, and
we no longer require this putting into relief of its con-
necting marks, our attention is no longer attracted to
the new and surprising, and we cease to speak of cause
and effect. Heat is said to be the cause of the tension
of steam ; but when the phenomenon becomes familiar

484 THE SCIENCE OF MECHANICS.

we think of the steam at once with the tension proper
to its temperature. Acid is said to be the cause of the
reddening of tincture of litmus \ but later we think of
the reddening as a property of the acid.
Hume, Hume first propounded the question, How can a

Kanti snd

schopen- thing A act on another thing -^ ? Hume, in fact, re-

haucr's eX"

pianations jects causalitv and recognises only a wonted succes-

of cause

and effect, sion in time. Kant correctly remarked that a necessary
connection between A and B could not be disclosed by
simple observation. He assumes an innate idea or
category of the mind, a Verstandesbegriff^ under which
the cases of experience are subsumed. Schopenhauer,
who adopts substantially the same position, distin-
guishes four forms of the '* principle of sufHcient rea-
son"— the logical, physical, and mathematical form,
and the law of motivation. But these forms differ only
as regards the matter to which they are applied, which
may belong either to outward or inward experience.
Cause and The natural and common-sense explanation is ap-
economical parcutly this. The ideas of cause and effect originally
oTSio?ght.^ sprang from an endeavor to reproduce facts in thought.
At first, the connection of A and By of C and Z>, of E
and Fy and so forth, is regarded as familiar. But after
a greater range of experience is acquired and a con-
nection between M and N is observed, it often turns
out that we recognise M as made up of Ay C, Ey and N
of By Dy Fy \\\^ counection of which was before a fa-
miliar fact and accordingly possesses with us a higher
authority. This explains why a person of experience
regards a new event with different eyes than the nov-
ice. The new experience is illuminated by the mass
of old experience. As a fact, then, there really does
exist in the mind an **idea'* under which fresh experi-
ences are subsumed ; but that idea has itself been de-

FORMAL DEVELOPMENT. 485

veloped from experience. The notion of the necessity
of the causal connection is probably created by our
voluntary movements in the world and by the changes
which these indirectly produce, as Hume supposed but
Schopenhauer contested. Much of the authority of
the ideas of cause and effect is due to the fact that they
are developed instinctively and involuntarily, and that
we are distinctly sensible of having personally con-
tributed nothing to their formation. We may, indeed,
say, that our sense of causality is not acquired by the
individual, but has been perfected in the develop-
ment of the race. Cause and effect, therefore, are
things of thought, having an economical office. It can-
not be said why they arise. For it is precisely by the
abstraction of uniformities that we know the question
"why." (See Appendix, V.)

4. In the details of science, its economical character Econom-

, , ical fea*

is still more apparent. The so-called descriptive sci- tures of

. . all laws of

ences must chiefly remain content with reconstructing nature,
individual facts. Where it is possible, the common fea-
tures of many facts are once for all placed in relief. But
in sciences that are more highly developed, rules for the
reconstruction of great numbers of facts may be embod-
ied in a single expression. Thus, instead of noting indi-
vidual cases of light- refraction, we can mentally recon-
struct all present and future cases, if we know that the
incident ray, the refracted ray, and the perpendicular
lie in the same plane and that sin or/sin y^= «. Here,
instead of the numberless cases of refraction in different
combinations of matter and under all different angles
of incidence, we have simply to note the rule above
stated and the values of «, — which is much easier. The
economical purpose is here unmistakable. In nature
there is no law of refraction, only different cases of re-

486 THE SCIEXCE OF MECI/AXICS.

fraction. The law of refraction is a concise compen-
dious rule, devised by us for the mental reconstruction
of a fact, and only for its reconstruction in part, that
is, on its geometrical side.
The econ- 5. The sciences most highly developed economically
mathemat- are those whose facts are reducible to a few numerable

* ■ *
ICSl SCI"

ences. elements of like nature. Such is the science of mechan-
ics, in which we deal exclusively with spaces, times,
and masses. The whole previously established econ-
omy of mathematics stands these sciences in stead.
Mathematics may be defined as the economy of count-
ing. Numbers are arrangement-signs which, for the
sake of perspicuity and economy, are themselves ar-
ranged in a simple system. Numerical operations, it
is found, are independent of the kind of objects operated
on, and are consequently mastered once for all. When,
for the first time, I have occasion to add five objects to
seven others, I count the whole collection through, at
once ; but when I afterwards discover that I can start
counting from 5, I save myself part of the trouble ;
and still later, remembering that 5 and 7 always count
up to 12, I dispense with the numeration entirely.
Arithmetic The objcct of all arithmetical operations is to save
andaiRc- ^j^^^.^ numeration, by utilising the results of our old
operations of counting. Our endeavor is, having done
a sum once, to preserve the answer for future use. The
first four rules of arithmetic well illustrate this view.
Such, too, is the purpose of algebra, which, substitut-
ing relations for values, symbolises and definitively
fixes all numerical operations that follow the same rule.
For example, we learn from the equation

FORMAL DEVELOPMENT. 487

that the more complicated numerical operation at the
left may always be replaced by the simpler one at the
right, whatever numbers x and y stand for. We thus
save ourselves the labor of performing in future cases
the more complicated operation. Mathematics is the
method of replacing in the most comprehensive and
economical manner possible, new numerical operations
by old ones done already with known results. It may
happen in this procedure that the results of operations
are employed which were originally performed centu-
ries ago.

Often operations involving intense mental effort The theory
may be replaced by the action of semi-mechanical minams.
routine, with great saving of time and avoidance of
fatigue. For example, the theory of determinants
owes its origin to the remark, that it is not necessary
to solve each time anew equations of the form

from which result

^1 •^ + ^1 ^^4- ^1

— 0

«2 •^+ ^2>'+^2
if

= 0,

_ ^1 ^2 ^2 ^X
a, ^2—^2^1

a^ ^2 ^2 ^1
^1^2 ^2^^!

but that the solution may be effected by means of the
coefficients, by writing down the coefficients according
to a prescribed scheme and operating with them me-
chanically. Thus,

a,b,
^3 ^2

— «i ^2 — ^2 ^

and similarly

^1 ^1
i^2 ^2!

— P, and ^^ ^'
''2 ^2

= N

Q-

488 THE SCIENCE OF MECHANICS.

Calculating Even a ioial disburdening of the mind can be ef-

machines.

fected in mathematical operations. This happens where
operations of counting hitherto performed are symbol-
ised by mechanical operations with signs, and our brain
energy, instead of being wasted on the repetition of
old operations, is spared for more important tasks.
The merchant pursues a like economy, when, instead
of directly handling his bales of goods, he operates
with bills of lading or assignments of them. The
drudgery of computation may even be relegated to a
machine. Several different types of calculating ma-
chines are actually in practical use. The earliest of
these (of any complexity) was the difference-engine of
Babbage, who was familiar with the ideas here pre-
sented.
Other ab- A numerical result is not always reached by the

methods of actual solution of the problem ; it may also be reached
results. indirectly. It is easy to ascertain, for example, that a
curve whose quadrature for the abscissa x has the value
x*"y gives an increment mx"'~^dx of the quadrature for
the increment dx of the abscissa.* But we then also know
that rmx*^''^dx = jr*"; that is, we recognise the quan-
tity x^ from the increment mx*^~'dx diS unmistakably
as we recognise a fruit by its rind. Results of this
kind, accidentally found by simple inversion, or by
processes more or less analogous, are very extensively
employed in mathematics.

That scientific work should be more useful the more
it has been used, while mechanical work is expended in
use, may seem strange to us. When a person who
daily takes the same walk accidentally finds a shorter
cut, and thereafter, remembering that it is shorter, al-
ways goes that way, he undoubtedly saves himself the
difference of the work. But memory is really not work.

FORMAL DEVELOPMENT, 489

It only places at our disposal energy within our present
or future possession, which the circumstance of igno-
rance prevented us from availing ourselves of. This
is precisely the case with the application of scientific
ideas.

The mathematician who pursues his studies with- Nocewity

of clear

out clear views of this matter, must often have the views on

this sub-
uncomfortable feeling that his paper and pencil sur- ject.

pass him in intelligence. Mathematics, thus pursued
as an object of instruction, is scarcely of more educa-
tional value than busying oneself with the Cabala. On
the contrary, it induces a tendency toward mystery,
which is pretty sure to bear its fruits.

6. The science of physics also furnishes examples Examples
of this economy of thought, altogether similar to those omy of °
we have just examined. A brief reference here will suf- phyafcs.
fice. The moment of inertia saves us the separate con-
sideration of the individual particles of masses. By
the force-function we dispense with the separate in-
vestigation of individual force-components. The sim-
plicity of reasonings involving force-functions springs
from the fact that a great amount of mental work had
tD be performed before the discovery of the properties
of the force-functions was possible. Gauss's dioptrics
dispenses us from the separate consideration of the
single refracting surfaces of a dioptrical system and
substitutes for it the principal and nodal points. But
a careful consideration of the single surfaces had to
precede the discovery of the principal and nodal points.
Gauss's dioptrics simply saves us the necessity of often
repeating this consideration.

We must admit, therefore, that there is no result of
science which in point of principle could not have been
arrived at wholly without methods. But, as a matter

490 THE SCIENCE OF MECHANICS.

Science a of fact, withiti the short span of a human life and with
problem, man's limited powers of memory, any stock of knowl-
edge worthy of the name is unattainable except by the
greatest mental economy. Science itself, therefore,
may be regarded as a minimal problem, consisting of
the completest possible presentment of facts with the
least possible expenditure of thought.

7. The function of science, as we take it, is to re-
place experience. Thus, on the one hand, science
must remain in the province of experience, but, on the
other, must hasten beyond it, constantly expecting con-
firmation, constantly expecting the reverse. Where
neither confirmation nor refutation is possible, science
is not concerned. Science acts and only acts in the
domain of uncompleted experience. Exemplars of such
branches of science are the theories of elasticity and
of the conduction of heat, both of which ascribe to the
smallest particles of matter only such properties as ob-
servation supplies in the study of the larger portions.
The comparison of theory and experience may be far-
ther and farther extended, as our means of observation
increase in refinement.

The princi- Experience alone, without the ideas that are asso-

ple of con- ..... 1 1 r

tinuity. the ciated With it, would torever remain strange to us.

entific Those ideas that hold good throughout the widest do-
mains of research and that supplement the greatest
amount of experience, are the most scientific. The prin-
ciple of continuity, the use of which everywhere per-
vades modern inquiry, simply prescribes a mode of
conception which conduces in the highest degree to the
economy of thought.

8. If a long elastic rod be fastened in a vise, the
rod may be made to execute slow vibrations. These
are directly observable, can be seen, touched, and

method.

FORMAL DEVELOPMENT. 491

graphically recorded. If the rod be shortened, the Example n<
vibrations will increase in rapidity and cannot be di- of the
rectly seen ; the rod will present to the sight a blurred science,
image. This is a new phenomenon. But the sensa-
tion of touch is still like that of the previous case ; we
can still make the rod record its movements ; and if
we mentally retain the conception of vibrations, we can
still anticipate the results of experiments. On further
shortening the rod the sensation of touch is altered ;
the rod begins to sound ; again a new phenomenon is
presented. But the phenomena do not all change at
once; only this or that phenomenon changes; conse-
quently the accompanying notion of vibration, which
is not confined to any single one, is still serviceable,
still economical. Even when the sound has reached
so high a pitch and the vibrations have become so
small that the previous means of observation are not
of avail, we still advantageously imagine the sounding
rod to perform vibrations, and can predict the vibra-
tions of the dark lines in the spectrum of the polarised
light of a rod of glass. If on the rod being further
shortened aii the phenomena suddenly passed into new
phenomena, the conception of vibration would no
longer be serviceable because it would no longer afford
us a means of supplementing the new experiences by
the previous ones.

When we mentally add to those actions of a human
being which we can perceive, sensations and ideas like
our own which we cannot perceive, the object of the
idea we so form is economical. The idea makes ex-
perience intelligible to us ; it supplements and sup-
plants experience. This idea is not regarded as a great
scientific discovery, only because its formation is so
natural that every child conceives it. Now, this is

492 THE SCIENCE OF MECHANICS.

exactly what we do when we imagine a moving body
which has just disappeared behind a pillar, or a comet
at the moment invisible, as continuing its motion and
retaining its previously observed properties. We do
this that we may not be surprised by its reappearance.
We fill out the gaps in experience by the ideas that
experience suggests.
All scien- 9- Yet not all the prevalent scientific theories origi-

ories not nated so naturally and artlessly. Thus, chemical, elec-
the'princ?" trical, and optical phenomena are explained by atoms.
8nu?ty?°° But the mental artifice atom was not formed by the
principle of continuity ; on the contrary, it is a pro-
duct especially devised for the purpose in view. Atoms
cannot be perceived by the senses ; like all substances,
they are things of thought. Furthermore, the atoms
are invested with properties that absolutely contradict
the attributes hitherto observed in bodies. However
well fitted atomic theories may be to reproduce certain
groups of facts, the physical inquirer who has laid to
heart Newton's rules will only admit those theories as
provisional helps, and will strive to attain, in some more
natural way, a satisfactory substitute.
Atoms and The atomic theory plays a part in physics similar
tai artifices, to that of Certain auxiliary concepts in mathematics ;
it is a mathematical model for facilitating the mental
reproduction of facts. Although we represent vibra-
tions by the harmonic formula, the phenomena of cool-
ing by exponentis^ls, falls by squares of times, etc. , no '
one will fancy that vibrations in themselves have any-
thing to do with the circular functions, or the motion
of falling bodies with squares. It has simply been ob-
served that the relations between the quantities inves-
tigated were similar to certain relations obtaining be-
tween familiar mathematical functions, and these more

FORMAL DEVELOPMENT. 493

familiar ideas are employed as an easy means of sup-
plementing experience. Natural phenomena whose re-
lations are not similar to those of functions with which
we are familiar, are at present very difiBcult to recon-
struct. But the progress of mathematics may facilitate
the matter.

As mathematical helps of this kind, spaces of more Nfaia-
than three dimensions may be used, as I have else- sioned

, , ^ . . , , spaces.

where shown. But it is not necessary to regard these,
on this account, as anything more than mental arti-
fices. *

* As the outcome of the labors of Lobatschewsky, Bolyai, Gauss, and Rie-
mann, the view has gradually obtained currency in the mathematical world,
that that which we call s^ct is a particular^ actual case of a more general^
conceivable case of multiple quantitative manifoldness. The space of sight
and touch is a threefold manifoldness; it possesses three dimensions ; and
every point in it can be defined by three distinct and independent data. But
it is possible to conceive of a quadruple or even multiple space-like manifold-
ness. And the character of the manifoldness may also be differently conceived
from the manifoldness of actual space. We regard this discovery, which is
chiefly due to the labors of Riemann, as a very important one. The properties
of actual space are here directly exhibited as objects of experience^ and the
pseudo-theories of geometry that seek to excogitate these properties by meta-
physical arguments are overthrown.

A thinking being is supposed to live in the surface of a sphere, with no
other kind of space to institute comparisons with. His space will appear to
him similarly constituted throughout. He might regard ^t as infinite, and
could only be convinced of the contrary by experience. Starting from any two
points of a great circle of the sphere and proceeding at right angles thereto on
other great circles, he could hardly expect that the circles last mentioned
would intersect. So, also, with respect to the space in which we live, only ex-
perience can decide whether it is finite, whether parallel lines intersect in it,
or the like. The significance of this elucidation can scarcely be overrated.
An enlightenment similar to that which Riemann inaugurated in science was
produced in the mind of humanity at large, as regards the surface of the earth,
by the discoveries of the first circumnavigators.

The theoretical investigation of the mathematical possibilities above re-
ferred to, has, primarily, nothing to do with the question whether things really
exist which correspond to these possibilities ; and we must not hold mathe-
maticians responsible for the popular absurdities which their investigations
have given rise to. The space of sight and touch is /ArM-dimensional ; that,
no one ever yet doubted. If, now, it should be found that bodies vanish from
this space, or new bodies get into it, the question might scientifically be dis-
cussed whether it would facilitate and promote our insight into things to con-
ceive experiential space as part of a four-dimensional or multi-dimensional

494 THE SCIE.VCE OF MECHANICS,

Hypotheses This is the case, too, with ail hypothesis formed
for the explanation of new phenomena. Our concep-
tions of electricity fit in at once with the electrical phe-
nomena, and take almost spontaneously the familiar
course, the moment we note that things take place as
if attracting and repelling fluids moved on the surface
of the conductors. But these mental expedients have
nothing whatever to do with the phenomenon itself,

space. Yet in such a case, this fourth dimension would, none the less, remain
a pure thing of thought, a mental fiction.

But this is not the way matters stand. The phenomena mentioned were
not forthcoming until a/ter the new views were published, and were then ex-
hibited in the presence of certain persons at spiritualistic stances. The fourth
dimension was a very opportune discover*- for the spiritualists and for theo-
logians who were in a quandary about the location of hell. The use the spiri-
tualist, makes of the fourth dimension is this. It is possible to move out of a
finite straight line, without passing the extremities, through the second dimen-
sion ; out of a finite closed surface through the third ; and, analogously, oat
of a finite closed space, without passing through the enclosing boundaries,
through the fourth dimension. Even the tricks that prestidigitateurs, in the
old days, harmlessly executed in three dimensions, are now invested with a
new halo by the fourth. But the tricks of the spiritualists, the tying or untying
of knots in endless strings, the removing of bodies from closed spaces, are all
performed in cases where there is absolutely nothing at stake. AH is purpose-
less jugglery. We nave not yet found an a<-a?M<:Arwr who has accomplished
parturition through the fourth dimension. If we should, the question would
at once become a serious one. Professor Simony's beautiful tricks in rope-
tying, which, as the performance of a prestidigitateur, are very admirable,
speak against, not for, the spiritualists.

Everyone is free to set up an opinion and to adduce proofs in support of
it. Whether, though, a scientist !:ha11 find it worth his while to enter into
serious investigations of opinions so advanced, is a question which his reason
and instinct alone can decide. If these things, in the end, should turn out to
be true, I shall not be ashamed of being the last to believe them. What I have
seen of them was not calculated to make me less sceptical.

I myself regarded multi-dimensioned space as a mathematico-physical

help even prior to the appearance of Riemann's memoir. Hut I trust that

no one will employ what I have thought, said, and written on this subject as a

basis for the fabrication of ghost stories. (Compare Mach, Die Ctsckickte und

die Wurzel dee Satzee von der ErkaUung der Arbeit.)

CHAPTER V.

THE RELATIONS OF MECHANICS TO OTHER DE- !

PARTMENTS OF KNOWLEDGE.

THE RELATIONS OF MECHANICS TO PHYSICS.

1 . Purely mechanical phenomena do not exist. The Tj^e events

•' * ^ ^ ^ of nature

production of mutual accelerations in masses is, to all<*ono*"-

* ' cUistvely

appearances, a purely dynamical phenomenon. But belong to
with these dynamical results are always associated cn<=«-
thermal, magnetic, electrical, and chemical phenom-
ena, and the former are always modified in proportion
as the latter are asserted. On the other hand, thermal,
magnetic, electrical, and chemical conditions also can
produce motions. Purely mechanical phenomena, ac-
cordingly, are abstractions, made, either intentionally
or from necessity, for facilitating our comprehension of
things. The same thing is true of the other classes of
physical phenomena. Every event belongs, in a strict
sense, to all the departments of physics, the latter be-
ing separated only by an artificial classification, which
is partly conventional, partly physiological, and partly
historical.

2. The view that makes mechanics the basis of the
remaining branches of physics, and explains all physical
phenomena by mechanical ideas, is in our judgment a
prejudice. Knowledge which is historically first, is
not necessarily the foundation of all that is subsequently

world.

496 THE SCIENCE OF MECHANICS,

The me- gained. As more and more facts are discovered and
aspects of classified, entirely new ideas of general scope can be

nature not

necessarily formed. We have no means of knowmg, as yet, w^hich

its funda-
mental of the physical phenomena go deepest^ whether the

mechanical phenomena are perhaps not the most super-
ficial of ^all, or whether all do not go equally deep. Even
in mechanics we no longer regard the oldest laiv, the
law of the lever, as the foundation of all the other
principles.
Artificiality The mechanical theory of nature, is, undoubtedly,

of the me- . .

chanicai in an historical view, both intelligible and pardonable ;

conception . i. , ,

of the and it may also, for a time, have been of much value.
But, upon the whole, it is an artificial conception.
Faithful adherence to the method that led the greatest
investigators of nature, Galileo, Newton, Sadi Carnot,
Faraday, and J. R. Mayer, to their great results, re-
stricts physics to the expression of actual facts^ and
forbids the construction of hypotheses behind the facts,
where nothing tangible and verifiable is found. If this
is done, only the simple connection of the motions of
masses, of changes of temperature, of changes in the
values of the potential function, of chemical changes,
and so forth is to be ascertained, and nothing is to be
imagined along with these elements except the physical
attributes or characteristics directly or indirectly given
by observation.

This idea was elsewhere * developed by the author
with respect to the phenomena of heat, and indicated,
in the same place, with respect to electricity. All hy-
potheses of fluids or media are eliminated from the
theory of electricity as entirely superfluous, when we
reflect that electrical conditions are all given by the

* Mach, Die Gesckiehte und dU Wnrul d€t S»tzes von der Erhaitun^ dtr
Arbeit.

ITS RELATIONS TO OTHER SCIENCES. 497

values of the potential function V and the dielectric Science
constants. If we assume the differences of the values based on

facts, not

of VX,o be measured (on the electrometer) by the forces, on hypoth-
and regard V and not the quantity of electricity Q as
the primary notion, or measurable physical attribute,
we shall have, for any simple insulator, for our quan-
tity of electricity

<?= 4;rJ \dx-+-dy-+-d^r^

(where x, y, z denote the coordinates and dv the ele-
ment of volume,) and for our potential*

%7t J \dx'^^ dy"^^ dz^

Here Q and ^appear as derived notions, in which no
conception of fluid or medium is contained. If we
work over in a similar manner the entire domain of
physics, we shall restrict ourselves wholly to the quan-
titative conceptual expression of actual facts. All su-
perfluous and futile notions are eliminated, and the
imaginary problems to which they have given rise fore-
stalled.

The removal of notions whose foundations are his-
torical, conventional, or accidental, can best be fur-
thered by a comparison of the conceptions obtaining
in the different departments, and by finding for the
conceptions of every department the corresponding
conceptions of others. We discover, thus, that tem-
peratures and potential functions correspond to the
velocities of mass-motions. A single velocity-value, a
single temperature-value, or a single value of potential
function, never changes alone. But whilst in the case
of velocities and potential functions, so far as we yet

* Using the terminology of Clausius.

498

THE SCIENCE OF MECHAXICS.

Desirabil-
ity of a
compara-
tive phys-
ics.

Circum-
stances
which fa-
vored the
develop-
ment of the
mechanical
view.

know, only differences come into consideration, the
significance of temperature is not only contained in its
difference with respect to other temperatures. Thermal
capacities correspond to masses, the potential of an
electric charge to quantity of heat, quantity of elec-
tricity to entropy, and so on. The pursuit of such re-
semblances and differences lays the foundation of a
comparative physics^ which shall ultimately render pos-
sible the concise expression of extensive groups of facts,
without arbitrary additions. We shall then possess a
homogeneous physics, unmingled with artificial atomic
theories.

It will also be perceived, that a real economy of
scientific thought cannot be attained by mechanical
hypotheses. Even if an hypothesis were fully com-
petent to reproduce a given department of natural phe-
nomena, say, the phenomena of heat, we should, by
accepting it, only substitute for the actual relations be-
tween the mechanical and thermal processes, the hy-
pothesis. ' The real fundamental facts are replaced by
an equally large number of hypotheses, which is cer-
tainly no gain. Once an hypothesis has facilitated,
as best it can, our view of new facts, by the substitu-
tion of more familiar ideas, its powers are exhausted.
We err when we expect more enlightenment from- an
hypothesis than from the facts themselves.

^3. The development of the mechanical view was
favored by many circumstances. In the first place, a
connection of all natural events with mechanical pro-
cesses is unmistakable, and it is natural, therefore, that
we should be led to explain less known phenomena by
better known mechanical events. Then again, it was
first in the department of mechanics that laws of gen-
eral and extensive scope were discovered^ A law of

ITS RELATIONS TO OTHER SCIENCES. 499

this kind is the principle of vis znva 2 (U"^ — C/q) =
2^pt (?'2 — p2^^ which states that the increase of the
vis vi7'a of a system in its passage from one position to
another is equal to the increment of the force-function,
or work, which is expressed as a function of the final
and initial positions. If we fix our attention on the
work a system can perform and call it with Helmholtz
the Spannkraft^ S,* then the work actually performed^
U, will appear as a diminution of the Spannkra/t, Ky
initially present; accordingly, S=K — Uy and the
principle of vis viva takes the form

2S -\- ^2mv^ = const y

that is to say, every diminution of the Spannkraft, is The cpn-
compensated for by an increase of the vis tnva. In this Energy,
form the principle is also called the law of the Conser-
vation of Energy y in that the sum of the Spannkra/t (the
potential energy) and the vis viva (the kinetic energy)
remains constant in the system. But since, in nature,
it is possible that not only vis viva should appear as the
consequence of work performed, but also quantities of
heat, or the potential of an electric charge, and so forth,
scientists saw in this law the expression of a mechanical
action as the basis of all natural actions. However,
nothing is contained in the expression but the fact of
an invariable quantitative connection between mechani-
cal and other kinds of phenomena.

4. It would be a mistake to suppose that a wide
and extensive view of things was first introduced into
physical science by mechanics. On the contrary, this

* Helmholts used this term in 1847; but it is not found in his subsequent
papers; and in 1882 {WUsentcka/tlichg Abkandlungen, II, 965) he expressly
discards it in favor of the English " potential energy." He even (p. 968) pre-
fers Clausius's word Ergal to S^nnkraft^ which is quite out of agreement
with modern terminology.— TVaffj.

500 THE SCIENCE OF MECHANICS,

Compre- insight was possessed at all times by the foremost

hensive- . . , , • , . ^

nessof inquirers and even entered into the construction of
condition, mechanics itself, and was, accordingly, not first created
suit, of me- by the latter. Galileo and Huygens constantly alter-

chfiinics. .

nated the consideration of particular details with the
consideration of universal aspects, and reached their
results only by a persistent effort after a simple and
consistent view. The fact that the velocities of indi-
vidual bodies and systems are dependent on the spaces
descended through, was perceived by Galileo and
Huygens only by a very detailed investigation of the
motion of descent in particular cases, combined with
the consideration of the circumstance that bodies gen-
erally, of their own accord, only sink. Huygens
especially speaks, on the occasion of this inquiry, of
the impossibility of a mechanical perpetual motion ;
he possessed, therefore, the modern point of view. He
felt the incompatibility of the idea of a perpetual motion
with the notions of the natural mechanical processes
with which he was familiar.
Exempiifi- Take the fictions of Stevinus — say, that of the end-
this in ste- less chaiu on the prism. Here, too, a deep, broad
searches, insight is displayed. We have here a mind, disciplined
by a multitude of experiences, brought to bear on an
individual case. The moving endless chain is to Ste-
vinus a motion of descent that is not a descent, a mo-
tion without a purpose, an intentional act that does
not answer to the intention, an endeavor for a change
which does not produce the change. If motion, gener-
ally, is the result of descent, then in the particular case
descent is the result of motion. It is a sense of the
mutual interdependence of v and h in the equation
V = \/ 2g/i that is here displayed, though of course in
not so definite a form. A contradiction exists in this

ITS RELATIONS TO OTHER SCIENCES. 501

fiction for Stevinus's exquisite investigative sense that
would escape less profound thinkers.

This same breadth of view, which alternates the Also, in the
individual with the universal, is also displayed, only in of carnot
this instance not restricted to mechanics, in the per- Mayer,
formances of Sadi Carnot. When Carnot finds that
the quantity of heat Q which, for a given amount of
work Z, has flgwed from a higher temperature / to a
lower temperature /*, can only depend on the tempera-
tures and not on the material constitution of the bodies,
he reasons in exact conformity with the method of
Galileo. Similarly does J. R. Mayer proceed in the
enunciation of the principle of the equivalence of heat
and work. In this achievement the mechanical view
was quite remote from Mayer's mind ; nor had he need
of it. They who require the crutch of the mechanical
philosophy to understand the doctrine of the equiva-
lence of heat and work, have only half comprehended
the progress which it signalises. Yet, high as we may
place Mayer's original achievement, it is not on that
account necessary to depreciate the merits of the pro-
fessional physicists Joule, Helmholtz, Clausius, and
Thomson, who have done very much, perhaps all, to-
wards the detailed establishment and perfection of the
new view. The assumption of a plagiarism of Mayer's
ideas is in our opinion gratuitous. They who advance
it, are under the obligation Xo prove it. The repeated
appearance of the same idea is not new in history. We
shall not take up here the discussion of purely personal
questions, which thirty years from now will no longer
interest students. But it is unfair, from a pretense of
justice, to insult men, who if they had accomplished
but a third of their actual services to science, would
have lived highly honored and unmolested lives.

502 THE SCIENCE OF MECHANICS.

The inter- 5. We shall now attempt to show that the broad

enceofthe vicw expressed in the principle of the conservation

faicts of nsi-

ture. of energy, is not peculiar to mechanics, but is a condi-

tion of logical and sound scientific thought generally.
The business of physical science is the reconstruction
of facts in thought, or the abstract quantitative expres-
sion of facts. The rules which we form for these recon-
structions are the laws of nature. In the conviction that
such rules are possible lies the law of causality. The
law of causality simply asserts that the phenomena of
nature are dependent on one another. The special em-
phasis put on space and time in the expression of the
law of causality is unnecessary, since the relations of
space and time themselves implicitly express that phe-
nomena are dependent on one another.

The laws of nature are equations between the meas>
urable elements afiyd , . . . a? of phenomena. As na-
ture is variable, the number of these equations is al-
ways less than the number of the elements.

If we know all the values ol afiyd , . .yhy which,
for example, the values of X/iv . . . are given, we may-
call the group afiyd. . . the cause and the group
XjjLV , . . the effect. In this sense we may say that the
effect is uniquely determined by the cause. The prin-
ciple of sufficient reason, in the form, for instance, in
which Archimedes employed it in the development of
the laws of the lever, consequently asserts nothing
more than that the effect cannot by any given set of
circumstances be at once determined and undetermined.

If two circumstances a and X are connected, then,
supposing all others are constant, a change of X will
be accompanied by a change of a, and as a general
rule a change of or by a change of X, The constant
observance of this mutual interdependence is met with

ITS RELATIONS TO OTHER SCIENCES. 503

in Stevinus, Galileo, Huygens, and other great inquir- Sense of

. , . , this inter-

ers. The idea is also at the basis of the discovery of depend-

ence at the

r^i/w/^T- phenomena. Thus, a chan^^e in the volume of basis of aii

. . J great dis-

a gas due to a change of temperature is supplemented coveries.
by the counter-phenomenon of a change of tempera-
ture on an alteration of volume ; Seebeck's phenome-
non by Peltier's effect, and so forth.
Care must, of course, be exercised, in
such inversions, respecting the form
of the dependence. Figure 235 will
render clear how a perceptible altera-
tion of a may always be produced by
an alteration of A, but a change of A
not necessarily by a change of or. The relations be-
tween electromagnetic and induction phenomena, dis-
covered by Faraday, are a good instance of this truth.

If a set of circumstances aByd , . ., by which ay»"oa«

' / ■' forms of ex-

second set Xuv . . , is determined, be made to pass pj"?*"**" <>'

, , '^ \ *^ ^ this truth.

from its initial values to the terminal values a' ^'y
d\ . ., then Xjjlv, . . also will pass intoA'/iV. . .
If the first set be brought back to its initial state, also
the second set will be brought back to its initial state.
This is the meaning of the ** equivalence of cause and
effect," which Mayer again and again emphasizes.

If the first group suffer only periodical changes, the
second group also can suffer only periodical changes,
not continuous permanent ones. The fertile methods
of thought of Galileo, Huygens, S. Carnot, Mayer,
and their peers, are all reducible to the simple but sig-
nificant perception, thai purely periodical alterations of
one set of circumstances can only constitute the source of
similarly periodical alterations of a second set of circum-
stances, not of continuous and permanent alterations. Such
maxims, as "the effect is equivalent to the cause,"

504 THE SCIENCE OF MECHANICS.

"work cannot be created out of nothing," '*a per-
petual motion is impossible," are particular, less defi-
nite, and less evident forms of this perception, which
in itself is not especially concerned with mechanics, but
is a constituent of scientific thought generally. With
the perception of this truths any metaphysical mystic-
ism that may still adhere to the principle of the con-
servation of energy* is dissipated. (See Appendix, VI. )
Purpose of AH ideas of conservation, like the notion of sub-

the ideas of ...

conserva- Stance, have a solid foundation in the economy of

tion. •'

' thought. A mere unrelated change, without fixed point
of support, or reference, is not comprehensible, not
mentally reconstructible. We always inquire, accord-
ingly, what idea can be retained amid all variations as
permanenty what law prevails, what equation remains
fulfilled, what quantitative values remain constant ?
When we say the refractive index remains constant in
all cases of refraction, ^remains = 9-810 w in all cases
of the motion of heavy bodies, the energy remains con-
stant in every isolated system, all our assertions have
one and the same economical function, namely that of
facilitating our mental reconstruction of facts.

II.

THE RELATIONS OF MECHANICS TO PHYSIOLOGY.

Conditions !• All scicnce has its origin in the needs of life.

deleTop"* However minutely it may be subdivided by particular
vocations or by the restricted tempers and capacities of
those who foster it, each branch can attain its full and
best development only by a living connection with the
whole. Through such a union alone can it approach

♦ When we reflect that the principles of science are all abstractions that
presuppose ref*etitions of similar cases, the absurd applications of the law of
the conservation of forces to the universe as a whole fall to the ground.

ment of
science.

ITS RELATIONS TO OTHER SCIENCES, 505

its true maturity, and be insured against lop-sided and
monstrous growths.

The division of labor, the restriction of individual Confusion

of the

inquirers to limited provinces, the investigation of means and

aims of

those provinces as a life-work, are the fundamental science,
conditions of a fruitful development of science. Only
by such specialisation and restriction of work can the
economical instruments of thought requisite for the
mastery of a special field be perfected. But just here
lies a danger — the danger of our overestimating the in-
struments, with which we are so constantly employed,
or even of regarding them as the objective point of
science.

2. Now, such a state of affairs has, in our opinion, physics
actually been produced by the disproportionate formal mad?tL
development of physics. The majority of natural in- pSySoiogy.
quirers ascribe to the intellectual implements of physics,
to the concepts mass, force, atom, and so forth, whose
sole office is to revive economically arranged expe-
riences, a reality beyond and independent of thought.
Not only so, but it has even been held that these forces
and masses are the real objects of inquiry, and, if once
they were fully explored, all the rest would follow from
the equilibrium and motion of these masses. A person
who knew the world only through the theatre, if brought
behind the scenes and permitted to view the mechan-
ism of the stage's action, might possibly believe that
the real world also was in need of a machine-room, and
that if this were once thoroughly explored, we should
know all. Similarly, we, too, should beware lest the
intellectual machinery, employed in the representation
of the world on the stage of thought, be regarded as the
basis of the real world.

.3. A philosophy is involved in any correct view of

5o6 THE SCIENCE OF MECHANICS,

The at- the relations of special knowledge to the great body of

tempt to ex- ^^ t» -f

plain feel- knowledge at large, — a philosophy that must be de-

inf{s by

motions, manded of every special investigator. The lack of it
is asserted in the formulation of imaginary problems,
in the very enunciation of which, whether regarded as
soluble or insoluble, flagrant absurdity is involved.
Such an overestimation of physics, in contrast to physi-
ology, such a mistaken conception of the true relations
of the two sciences, is displayed in the inquiry whether
it is possible to explain feelings by the motions of
atoms?

Explication Let US Seek the conditions that could have impelled

of this . . ^

anomaly, the mind to formulate so curious a question. We find
in the first place that greater confidence is placed in our
experiences concerning relations of time and space ;
that we attribute to them a more objective, a more real
character than to our experiences of colors, sounds,
temperatures, and so forth. Yet, if we investigate the
matter accurately, we must surely admit that our sen-
sations of time and space are just as much sensations
as are our sensations of colors, sotinds, and odors, only
that in our knowledge of the former we are surer and
clearer than in that of the latter. Space and time are
well-ordered systems of sets of sensations. The quan-
tities stated in mechanical equations are simply ordinal
symbols, representing those members of these sets
that are to be mentally isolated and emphasised. The
equations express the form of interdependence of these
ordinal symbols.

A body is a relatively constant sum of touch and
sight sensations associated with the same space and
time sensations. Mechanical principles, like that, for
instance, of the mutually induced accelerations of two
masses, give, either directly or indirectly, only some

ITS RELATIONS TO OTHER SCIENCES. 507

combination of touch, sight, light, and time sensations.
They possess intelligible meaning only by virtue of
the sensations they involve, the contents of which may
of course be very complicated.

It would be equivalent, accordingly, to explaining Mode of

- . . , ,. avoiding

the more simple and immediate by the more compli- such er-

. rors.

cated and remote, if we were to attempt to derive sen-
sations from the motions of masses, wholly aside from
the consideration that the notions of mechanics are
economical implements or expedients perfected to
represent mechanical and not physiological or psycho-
logical facts. If the means and aims of research were
properly distinguished, and our expositions were re-
stricted to the presentation of actual fads, false prob-
lems of this kind could not arise.

4. All physical knowledge can only mentally repre- The princi-
sent and anticipate compounds of those elements we ch?n?csn^t
call sensations. It is concerned with the connection of don but
these elements. Such an element, say the heat of a body ll^ect o"
A, is connected, not only with other elements, say with
such whose aggregate makes up the flame B, but also
with the aggregate of certain elements of our body, say
with the aggregate of the elements of a nerve N. As
simple object and element Nis not essentially, but only
conventionally, different from A and B. The connection
of A and ^ is a problem of physics, that of A and N a
problem oi physiology. Neither is alone existent; both
exist at once. Only provisionally can we neglect
either. Processes, thus, that in appearance are purely
mechanical, are, in addition to their evident mechani-
cal features, always physiological, and, consequently,
also electrical, chemical, and so forth. The science of
mechanics does not comprise the foundations, no, nor
even a part of the world, but only an aspect of it.

APPENDIX.

(See page 140.}

In an exhaustive study in the Zeitschrift fiir Volker-
psychologies 1884, Vol. XIV, pp. 365-410, and Vol. XV,
pp. 70-135, 337-387, entitled Die Entdeckung des Be-
harrungsgesetzeSy E. Wohlwill has shown that the prede-
cessors and contemporaries of Galileo, nay, even Gali-
leo himself, only t^ery gradually abandoned the Aristo-
telian conceptions for the acceptance of the law of in-
ertia. Even in Galileo's mind uniform circular motion
and uniform horizontal motion occupy distinct places.
Wohlwill's researches are very acceptable and show
that Galileo had not attained perfect clearness in his
own new ideas and was liable to frequent reversion to
the old views, as might have been expected.

Indeed, from my own exposition the reader will
have inferred that the law of inertia did not possess
in Galileo's mind the degree of clearness and univer-
sality that it subsequently acquired. (See pp. 140
and 143.) With regard to my exposition at pages
1 40-141, however, I still believef in spite of the opin-
ions of Wohlwill and Poske, that I have indicated the
point which both for Galileo and his successors must
have placed in the most favorable light the transition B
from the old conception to the new.

5IO THE SCIENCE OF MECHANICS.

II.
(See page 218.}

H. Streintz*s objection {Die physikalischen Grund-
lagen der Mechaniky Leipsic, 1883, p. 117), that^ com-
parison of masses satisfying my definition can be ef-
fected only by astronomical means, I am unable to ad-
mit. The expositions on pages 202, 218-221 amply
refute this. Masses mutually produce in each other
accelerations in impact, when subject to electric and
magnetic forces, and when connected by a string on
Atwood's machine.

My definition is, the outcome of an endeavor to
establish the interdependence of phenomena and to re-
move all metaphj'sical obscurity, without accomplish-
ing on this account less than other definitions have
done. I have pursued exactly the same course with
respect to ideas ** quantity of electricity "( ^^/-^^r ///V
Grundbegriffe der Eiektrostatiky Vortrag gehaltcn auf der
inter nationalen eiektrischen Ausstellung, Vienna, Septem-
ber 4, 1883), '* temperature," ''quantity of heat" {Zett-
schrift fiir den physikalischen und chemischen Unterricht^
Berlin, 1888, No. I), and so forth.

III.

(See page 226.)

My views concerning physiological time, the sensa-
tion of time, and partly also concerning physical time,
I have expressed elsewhere (see Beit rage zur Analyse
der Empfindungeny Jena, Fischer, 1886, pp. 103-in,
166-168). As in the*study of thermal phenomena we
select as our measure of temperature an arbitrarily
chosen volume, which varies in almost parallel correspon-
dence with our sensation of heat, and which is not liable
to the uncontrollable disturbances of our organs of sen-

APPENDIX, 511

sation, so, for similar reasons^ we select, in this instance,
as our measure of time, an arbitrarily chosen mot ion j (the
wangle of the earth*s rotation, or path of a free body,)
which^jroceeds in almost parallel correspondence with
our sensation of time. Once we have made clear to our-
selves that we are concerned only with the ascertain-
ment of the interdependence of phenomena, as I pointed
out as early as 1 865 ( Ueber den Zeitsinn des Ohres, Sitzungs-
berichte der Wiener Akademie) and 1866 (Fichte's Zeit-
schrift fUr Philosophic^ all metaphysical obscurities dis-
appear. (Compare J. Epstein, Die logischen Principien
der Zeiimessungy Berlin, 1887.)

IV.
(See page 238.)

Of the treatises which have appeared since 1883 on
the law of inertia, all of which furnish welcome evidence
of a heightened interest in this question, I can here
only briefly mention that of Streintz {Physikalische
Grundlagen der Mechanik, Leipsic, 1883) and that of L.
Lange {Die gcschichtliche Entwicklung des Bewegungs-
begriffesy Leipsic, 1886).

The expression ** absolute motion of translation"
Streintz correctly pronounces as devoid of meaning and
consequently declares certain analytical deductions,
to which he refers, superfluous. On the other hand,
with respect to rotation^ Streintz accepts Newton's po-
sition, that absolute rotation can be distinguished from
relative rotation. In this point of view, therefore, one
can select every body not affected with absolute rota-
tion as a body of reference for the expression of the
law of inertia.

I cannot share this view. For me, only relative
motions exist {Er/ialtung der Arbeit^ p. 48 ; Science of

512 THE SCIENCE OF MECHANICS,

Mechanics, p. 229), and I can see, in this regard, no
distinction between rotation and translation. When a
body moves relatively to the fixed stars, centrifugal
forces are produced ; when it moves relatively to some
different body, and not relatively to the fixed stars, no
centrifugal forces are produced. I have no objection
to calling the first rotation ** absolute" rotation, if it
be remembered that nothing is meant by such a desig-
nation except relative rotation with respect to the fixed
stars. Can we fix Newton's bucket of water, rotate the
fixed stars, and then prove the absence of centrifugal
forces ?

The experiment is impossible, the idea is meaning-
less, for the two cases are not, in sense-perception,
distinguishable from each other. I accordingly regard
these two cases as thexaw^ case and Newton's dis-
tinction as an illusion {Science of Mechanics, p. 232).

But the statement is correct that it is possible to
find one's bearings in a balloon shrouded in fog, by
means of a body which does not rotate with respect to
the fixed stars. But this is nothing more than an in-
direct orientation with respect to the fixed stars ; it is
a mechanical, substituted for an optical, orientation.

I wish to add the following remarks in answer to
Streintz's criticism of my view. My opinion is not to
be confounded with that of Euler (Streintz, pp. 7, 50),
who, as Lange has clearly shown, never arrived at
any settled and intelligible opinion on the subject.
Again, I never assumed that remote masses only\ and not
near ones, determine the velocity of a body (Streintz, p.
7); I simply spoke of an influence independent of dis-
tance. In the light of my expositions at pages 222-
245, the unprejudiced and careful reader will scarcely
maintain with Streintz (p. 50), that after so long a pe-

APPENDIX, 513

riod of time, without a knowledge of Newton and
Euler, I have only been led to views which these in-
quirers long ago held, but were afterwards, partly by
them and partly by others, rejected. Even my re-
marks of 1872, which were all that Streintz knew, can-
not justify this criticism. These were, for good rea-
sons, concisely stated, but they are by no means so
meagre as they must appear to one who knows them
only from Streintz's criticism. The point of view
which Streintz occupies, I at that time expressly re-
jected.

Lange's treatise is, in my opinion, one of the best
that have been written on this subject. Its methodical
movement wins at once the reader's sympathy. Its
careful analysis, and study, from historical and critical
points of view, of the concept of motion, have pro-
duced, it seems to me, results of permanent value. I
also regard its clear emphasis and apt designation of
the principle of ''particular determination " as a point
of much merit, although the principle itself, as well as
its application, is not new. The principle is really at
the basis of all measurement. The choice of the unit
of measurement is convention ; the number of measure-
ment is a result of inquiry. Every natural inquirer who
is clearly conscious that his business is simply the in-
vestigation of the interdependence of phenomena, as I
formulated the point at issue a long time ago (1865-
1866), employs this principle. When, for example,
{Mechanics, p. 218 et seqq.), the negative inverse ratio
of the mutually induced accelerations of two bodies is
called the mass-ratio of these bodies, this is a conven-
tion, e3?J)ressly acknowledged as arbitrary ; but that
these ratios are independent of the kind and of the
order of combination of the bodies is 2l result of inquiry.

514 THE SCIENCE OF AfECHANICS.

I might adduce numerous similar instances from the
theories of heat and electricity as well as from other
provinces. Compare Appendix, II.

Taking it in its simplest and most perspicuous form,
the law of inertia, in Lange's view, would read as fol*
lows:

''Three material points P^^ P^j P^y are simulta-
"neously hurled from the same point in space and
** then left to themselves. The moment we are certain
''that the points are not situated in the same straight
"line, we join each separately with any fourth point in
"space, Q. These lines of junction, which we may
"respectively call G^, G^y G^, form, at their point of
" meeting, a three-faced solid angle. If now we make
"this solid angle preserve, with unaltered rigidity,
" its form, and constantly determine in such a manner
"its position, that P^ shall always move on the line

'* G^, P^on the line 6^2 » -^s ^^ *^^ ^^^® ^3» these lines
" may be regarded as the axis of a coordinate or iner-
" tial system, with respect to which every other ma-
"terial point, left to itself, will move in a straight line.
"The spaces described by the free points in the paths
"so determined will be proportional to one another."

A system of coordinates with respect to which three
material points move in a straight line is, according to
Lange, under the assumed limitations, a simple con-
vention. That with respect to such a system also a
fourth or other free material point will move in a
straight line, and that the paths of the different points
will all be proportional to one another, are results of
inquiry.

In the first place, we shall not dispute the fact that
the law of inertia can be referred to such a system of
time and space coordinates and expressed in this form.

APPENDIX, 515

Such an expression is less fit than Streintz's for prac-
tical purposes, but on the other hand, is, for its method-
ical advantages, more attractive. It especially appeals
to my mind, as a number of years ago I was engaged
with similar attempts, of which not the beginnings but
only a few remnants {Mechanics, pp. 234-235) are left.
I abandoned these attempts because I was convinced
that we only apparently evade by such expressions ref-
erences to the fixed stars and the angular rotation of
the earth. This, in my opinion, is also true of the
forms in which Streintz and Lange express the law.

In point of fact, it was precisely by the considera-
tion of the fixed stars and the rotation of the earth
that we arrived at a knowledge of the law of inertia as
it at present stands, and without these foundations we
should never have thought of the explanations here
discussed {Mechanics, 232-233). The consideration of
a small number of isolated points, to the exclusion of
the rest of the world, is in my judgment inadmissible
{Mechanics, pp. 229-235).

It is quite questionable, whether a fourth material
point, left to itself, would, with respect to Lange*s
'*inertial system,'* uniformly describe a straight line, if
the fixed stars were absent, or not invariable, or could
not be regarded with sufficient approximation as in-
variable.

The most natural point of view for the candid in-
quirer must still be, to regard the law of inertia pri-
marily as a tolerably accurate approximation, to refer
it, with respect to space, to the fixed stars, and, with
respect to time, to the rotation of the earth, and to
await the correction, or more precise definition, of our
knowledge from future experience, as I have explamed
on page 237 of this book.

5i6 THE SCIENCE OF MECHANICS, \

Upon the whole, the treatises that have appeared
since 1883 convince me that my expositions have not
j'et been fully considered, and I have therefore left the
text of this subject unaltered.

(See page 485.)

In the text I have employed the term *' cause" in
the sense in which it is ordinarily used. I may add
that with Dr. Carus,* following the practice of the
German philosophers, I distinguish ''cause," or ^^-a/-
grund, from Erkenntnissgrund. I also agree with Dr.
Carus in the statement that '* the signification of cause
and effect is to a great extent arbitrary and depends
much upon the proper tact of the observer." \

The notion of cause possesses significance only as
a means of provisional knowledge or orientation. In
any exact and profound investigation of an event the
inquirer must regard the phenomena as dependent on
one another in the same way that the geometer regards
the sides and angles of a triangle as dependent on one
another. He will constantly keep before his mind, in
this way, all the conditions of fact.

VI.
(See page 504.)

The principle of energy is only briefly treated in
the text, and I should like to add here a few remarks
on the following four treatises, discussing this subject,
which have appeared since 1883 : Die physikalischen
Grundsatze der elektrischen Kraftiibertragung, by J. Pop-

* See his Grnnd, Ursacke unti ZwecJi; R. v. Grurobkow, Dresden, iS8x,
and his Fundamrntal ProbUmx, pp. 79-91, Chicago : The Open Court Publish-
ing Co., 1891.

t Fundamental Problems, p. 84.

APPENDIX. 517

per, Vienna, 1883 ; Die Lehre von der Energie^ by G.
Helm, Leipsic, 1887 ; Das Princip der Erhaltung der
Energie, by M. Planck, Leipsic, 1887 ; and Das Pro-
blem der Continuitdt in der Mathematik und Mechanik,
by F. A. MuUer, Marburg, 1886.

The independent works of Popper and Helm are,
in the aim they pursue, in perfect accord, and they
quite agree in this respect with my own researches, so
much so in fact that I have seldom read anything that,
without the obliteration of individual differences, ap-
pealed in an equal degree to my mind. These two
authors especially meet in their attempt to enunciate
a general science of energetics ; and a suggestion of this
kind is also found in a note to my treatise, Ueber die
Erhaltung der Arbeit, page 54.

In 1872, in this same treatise (pp. 42 et seqq.), I
showed that our belief in the principle of excluded per-
petual motion is founded on a more general belief in
the unique determination of one group of (mechanical)
elements, afty . . ., by a group of different elements,
xyz . . . Planck's remarks at pages 99, 133, and 139
of his treatise, essentially agree with this ; they are
different only in form. Again, I have repeatedly re-
marked that all forms of the law of causality spring
from subjective impulses, which nature is by no means
compelled to satisfy. In this respect my conception is
allied to that of Popper and Helm.

Planck (pp. 21 et seqq., 135) and Helm (p. 25 et
seqq.) meiltion the "metaphysical" points of view by
which Mayer was controlled, and both remark (Planck,
p. 25 et seqq., and Helm, p. 28) that also Joule, though
there are no direct expressions to justify the conclusion,
must have been guided by similar ideas. To this last
I fully assent.

5X8 THE SCIENCE OF MECHANICS,

With respect to the so-called ** metaphysical''
points of view of Mayer, which, according to Helm-
holtz, are extolled by the devotees of metaphysical
speculation as Mayer's highest achievement, but which
appear to Helmholtz as the weakest feature of his ex-
positions, I have the following remarks to make. With
maxims, such as "Out of nothing, nothing comes/'
**The effect is equivalent to the cause," and so forth,
one can never convince another of anything. How little
such empty maxims, which until recently were admit-
ted in science, can accomplish, I have illustrated by
examples in my treatise Die Erhaltung der Arbeit, But
in Mayer's case these maxims are, in my judgment,
not weaknesses. On the contrary, they are with him
the expression of a powerful instinctive yearning, as yet
unsettled and unclarified, after a sound, substantial
conception of what is now called energy. This desire I
should not exactly call metaphysical. We now kno'w
that Mayer was not wanting in the conceptual power
to give to this desire clearness. Mayer's attitude in
this point was in no respect different from that of Gali-
leo, Black, Faraday, and other great inquirers, although
perhaps many were more taciturn and cautious than he.

I have touched upon this point before in the Bet-
trdge zur Analyse der Empfindungen^ Jena, 1886, p. 161
et seqq. Aside from the fact that I do not share the
Kantian point of view, in fact, occupy no metaphysical
point of view, not even that of Berkeley, as hasty
readers of my last-mentioned treatise have assumed,
I agree with F. A. Muller's remarks on this question
(p. 104 et seqq).

CHRONOLOGICAL TABLE

OF A FFW

EMINENT INQUIRERS

AND OF

THEIR MORE IMPORTANT MECHANICAL WORKS.

A.RCHIMBDBS (287-212 B. €.)• A Complete edition of his works was
published, with the commentaries of Eutocius, at Oxford, in
1792 ; a French translation by F. Peyrard (Paris, 1808): a Ger-
man translation by Ernst Nizze (Stralsund, 1824).

Leonardo da Vinci (1452-1519). Leonardo's scientific manuscripts
are substantially embodied in H. Grothe's work, "Leonardo
da Vinci als Ingenieur und Philosoph " (Berlin, 1874).

GuiDO Ubaldi(o) e Marchionibus Montis (1545-1607). Mecham-
corum Liber (Pesaro, 1577).

S. Stbvinus (1548-1620). Begkinsehn der Weegkonst (Leyden,
1585); Hypomnemata Matkematica (Leyden, 1608).

Galilbo (1564-1642). Discorsi e dimostrazioni matematicke (Ley-
den, 1638). The first complete edition of Galileo's writings
was published at Florence (i 842-1 856), in fifteen volumes 8vo.

Kbplbr (1571-1630). Astronomia Nova (Prague, 1609); Harmo-
nice Mundi (Linz, 1619); SUreometria Dolioriim (L.\m, 161 5).
Complete edition by Frisch (Frankfort, 1858).

Marcus Marci (1595-1667). De Proportiotu Afo/ us (Prsigue, 1639).

Dbscartes (1596-1650). Principia PAi/(?s0p/n\r (Xmsierdaim, 1644).

RoBBRVAL (1602-1675). Sur la composition des mouvements. Anc,
Mint, de rAcad. de Paris. T. VI.

Gubrickb (1602- 1686). Experimenta Nova, ut Vocantur Magde-
burgica (Amsterdam. 1672).

520 THE SCIENCE OF MECHANICS,

Fbrmat (1601-1665). Varia Opera (Toulouse. 1679).

ToRRicBixi (1608-1647). Opera Geonietrica (Florence, 1644).

Wazxis (16x6-1703). Mechanica Sive de Motu (London, 1670).

Mariottb (1620-1684). (Euvres (Leyden, 1717).

Pascal (1623-1662). R^cit de In grande experience de riquilibre
dts liqueurs (Paris, 1648); Traiti de riquilibre des liqueurs et
de la pesanteur de la masse de Vair. (Paris, 1662).

BoYLX (1627-1691). Experimenta Pkvsico Mechanica (London,
1660).

HUYGBNS (1629- 1695). A Summary Account of the Laws of Men
Hon. Philos. Trans. 1669; Horologium Oscillatorium (Paris,
1673); Opuscula Posthuma (Leyden, 1703).

Wrbn (1632- 1 723). Lex Naturce de Collisione Corporum, Philos.
Trans. 1669.

Lami (1640-17 1 5). Nouvelle manikre de dimontrer les principaux
thioremes des iUmens des micaniques (Paris, 1687).

Nbwton (1642 -1726). Philosophic Naturalis Principia Mathema-
tica (London, 1686).

Lbibnitz (1646-1716). Acta Eruditorum^ 1686. 1695 ; Leibnitzii
et Joh, Bernoullii Comer cium Epistolicum (Lausanne and Ge-
neva, 1745)-

Jambs Bbrnoulli (1654-1705). Opera Omnia ^Geneva, 1744).

Varignon (1654-Z722). Projet d^une nouvelle mecanique (Paris.
1687).

John Bbrnoulli (1667 -1748). Acta Erudit. 1693; Opera Omnia
(Lausanne, 1742).

Maupbrtuis (1698-1759). M^m. de VAcad. de Paris^ 1740 ; M^m.
de VAcad. de Berlin^ 1745. i747 \ (Euvres (Paris, 1752).

Maclaurin (1698-1746). A Complete System of Fluxions (Edin-
burgh, 1742).

Daniel Bbrnoulli (1700-1782). Comment. Acad. Petrop., T, I.
Hydrodynamica (Strassburg, 1738).

EuLER (1707-1783). Mechanica siz'e Mot us Scientia (Petersburg:,
1736) ; Methodus Inveniendi Lineas Curvas (Lausanne. 1744).

CHRONOLOGICAL TABLE, 521

Numerous articles in the volumes of the Berlin and St. Peters-
burg academies.

Clairaut (1713-1765). Thiorie de la figure de la terre (Paris,

1743).
D'Alembbrt (1717-1783). TraiU de dynamique (Paris, 1743).

Lagrange (1736-1813). Essai d^une nouifelle methode pour diter-
miner les maxima et minima. Misc. Taurin. 1762 ; Micanique
analytique (Paris, 1788).

Laplace (1749-1827). Mecanique cileste (Paris, 1799).

Fourier (1768-1830). Thiorie analytique de-la chaleur (Paris,
1822).

Gauss (1777-1855). De Figura Fluidorum in Statu y^qui/ihrii.
Comment. Societ. Gottin;^., 1828; Neues Princip det Mechanik
(Crelle's Journal, IV, 1829); Intensitas Vis Magnetica Terrestris
ad Mensuram Absolutam Revocata (1833). Complete works
(G5ttingen, 1863).

PoiNSOT (1777-1859). Elements de statique (Paris, 1804).

PoNCELBT (1788-1867). Cours de mecanique {Meiz, 1826).

Belanger (1790-1874). Cours de micanique (Paris, 1847).

MoBi us (179a 1867). 5/tf/i>& (Leipsic. 1837).

CoRiOLis (1792- 1843). Traiti de nUcanique (Paris, 1829).

C. G. J. Jacobi (1804-1851). Vorlesungen Uber Dynamik, heraus-
gegeben von Clebsch (Berlin, 1866).

W. R. Hamilton (1805-1865). Lectures on Quaternions, 1853. —
Essays.

Grassmann (1809- 1 877). Ausdehnungslehre (Leipsic, 1844).

INDEX.

Absolate, space, time. etc. (See the

nouns.)
Absolate units, 278, 284.
Abstractions, economical character

of, 483.
Acceleration, Galileo on, i3iiet seqq.;

Newton on, 238 ; also 218, 230, 236,

243. H5'
Action and reaction, Newton on, 198-

201, 242.

Action, least, principle of, 364-380,
454 ; sphere of, 383.

Adaptation, in nature,4S2; of thoughts
to facts, 6.

Adhesion plates, 1x5.

Aerostatics. (See air.)

Affined, 166.

Air, expansive force of isolated por-
tions of, 127 ; quantitative data of,
124; weight of, 113; pressure of,
114 et seqq.

Air-pump, experiments, 122 et seqq.;
the mercurial, 123.

Alcohol and water, mixture of, 384 et
seq.

Algebra, economy of, 486.

Algebraical mechanics, 4G6.

All, The, necessity of its considera-
tion in research, 23S1 461.

Analytical mechanics, 465-480.

Analytic method, 466.

Animal free in space, 290.

Animistic points of view in mechan-
ics, 461 et seq.

Archimedes, on the lever and the
centre of gravity, 8-1 1 ; critique of
his deduction, 13-14 ; illustration
of its value, 10; on hydrostatics,
86-88 ; various modes of deduction

of his hydrostatic principle, 104;
illustration of his principle, 106.

Areas, the law of the conservation of,
293-305.

Areometer, effect of particles sus-
pended in liquids on, 208.

Artifices, mental, 492 et seqq.

Assyrian monuments, i.

Atmosphere. (See Air.)

Atoms, mental artifices, 492.

Attraction, 246.

Atwood' s machine, 149.

Avenarius, R., ix.

Babbage, on calculating machines.
488.

Babo, von, apparatus of, 150.

Ballistic pendulum, 328.

Balls, elastic, symbolising pressures
in liquids, 419.

Bandbox, rotation of, 301.

Barometer, height of mountains de-
termined by, XI 5, X17.

Base, pressure of liquids on, 90, 99.

Belanger, on impulse, 271.

Berkeley, 518.

Bernoulli, Daniel, his geometrical
demonstration of the parallelo-
gram of forces, 40-42 ; criticism of
Bernoulli's demonstration, 42-46;
on the law of areas, 293 ; on the
principle of vis viva, 343, 348 ; on
the velocity of liquid efflux, 403; his
hydrodynamic principle,4o8; on the
parallelism of strata, 409; his dis-
tinction of hydrostatic and hydro-
dynamic pressure, 413.

Bernoulli, James, on the catenary,
74 ; on the centre of oscillation, 33X

524

THE SCIENCE OF MECHANICS.

e( seq.; on the brachistochrone,
426; on the isoperiinetrical prob-
lems, 428 et seq.; his character, 428 ;
his quarrel with John, 431 ; his Pro-
gramma, 430.

Bernoulli, John, his generalisation of
the principle of virtual velocities.
56; on the catenary, 74; on centre
of oscillation, 333 335; on the prin-
ciple of vit viva, 343 ; on the anal-
ogies between motions of masses
and light, 372; his liquid pendulum,
410 ; on the brachistochrone, 425 et
seqq.; his character, 427 ; his quar-
rel with James, 431 ; his solution of
the isoperimetrical problem, 431.

Black, his discovery of carbonic acid
gas, 124.

Boat in motion, Huygens's fiction of

a, 315. 325-
Body, definition of, 506.
Bolyai, 493.
Bomb, a bursting, 293.
Bouguer, on the figure of the earth,

395-

Boyle, his law, 125 et seq.; his inves-
tigations in aerostatics, 123.

Brachistochrone, problem of the, 423
et seqq.

Brahe, Tycho, on planetary motion,
187.

Bruno.Giordano, his martyrdom, 446.

Bubbles, 392.

Bucket of water, Newton's rotating,
227, 232, 512.

Cabala, 489.

Calculating machines, 488.

Calculus, difterential, 424 ; of varia-
tions, 436 et seqq.

Canal, fluid, equilibrium of, 396 et
seqq.

Cannon and projectile, motion of, 291.

Canton,on compressibility of liquids,
92.

Carnot. his performances, 501 ; his
formula, 327.

Carus, P., on cause, 516.

Catenary, The, 74, 379, 425.

Cauchy, 47.

Causality, 483 et seqq.; 502.

Cause and effect, economical char-
acter of the ideas, 483 : equivalence
off 502, 503; Mach on, 516; Cams
on, 516.

Causes, efficient and final, 368.

Cavendish, his discovery of hydro-
gen, 124.

Cells of the honeycomb, 453.

Centimetre-gramme-second system.
2B5.

Central, centrifugal, and centripetal
force. (See Force.)

Centre of gravity, 14 et seqq.; descent
of, 32 ; descent and ascent of, 174
et seqq., 408; the law of the con*
servation of the, 287-303.

Centre of gyration, 334.

Centre of oscillation, 173 et seqq., 331
-335 ; Mersenne, Descartes, and
Huygens on, 174 et seqq.; relations
of, to centre of gravity, 180-185;
convertibility of, with point of sus-
pension, 186.

Centre of percussion, 327.

Chain, Stevin's endless, 25 et seqq.,
500 ; motion of, on inclined plane.

347.

Change, unrelated, 504.

Character, an ideal universal, 481.

Chinese language, 482.

Church, conflict of science and, 446.

Circular motion, law of, x6o, 161.

Clairaut, on vis viva, work, etc., 348 ;
on the figure of the earth, 395 ; on
liquid equilibrium, 396 et seq.; on
level surfaces, etc., 398.

Classes and trades, the function of in
the development of science, 4.

Clausius, 497, 499, 301.

Coefficients, indeterminate, La-
grange' s, 471 et seq.

Collision of bodies. (See Impact

Colors, analysis of. 481.

Column, rest of a heavy, 258.

Commandinus, 87.

Communication, the economy of, 78.

Comparative physics, necessity of.

498.
Component of force, 34.
Composition of forces, see Forces *

Gauss's principle and the, 364.

INDEX.

525

Compression of liquids and |{a8cs,407.

Conrad us, Balthasar, 308.

Conservation, of energy, 499 et seq.,
516 et seqq.; of quantity of motion,
Descartes and Leibnits on, 372, 274;
purpose of the ideas of. 504.

Conservation of momentum, of the
centre of gravity, and of areas, laws
of the, 287-305 ; these laws, the ex-
pression of the laws of action and
reaction and inertia, 303.

Conservation of momentum and vis
viva interpreted, 326 et seq.

Constancy of quantity of matter, mo-
tion, and energy, theological basis
of, 456.

Constraint. 333, 352 ; least, principle

of. 350-364.
Continuity, the principle of, 140, 490

et seqq.
Continuum, physico-mechanical, 109.
Coordinates, forces a function of, 397

see Force-function.
Copernicus, 457, 232.
Coriolis, on vis viva and work, 272.
Counter-phenomena, 303.
Counter-work, 363, 366.
Counting, economy of, 486.
Courtivron, his law of equilibrium,

73.
Ctesibtus, his air-gun, iia

Currents, oceanic, 302.

Curtius Rufus, 210.

Curve-eleuients, variation of, 432.

Curves, roixima and minima of, 429.

Cycloids, X43, 186, 379, 427.

Cylinder, double, on a horizontal sur-
face, 60; rolling on an inclined
plane. 345.

Cylinders, axal, symbolising the rela-
tions of the centres of gravity and
oscillation, 183.

D'Alembert, his settlement of the
dispute concerning the measure of
force, 149, 276; bis principle, 331-

343-
D'Arcy, on the law of areas, 293.
Darwin, his theories, 452, 459.
Declination from free motion, 352-

356.

Deductive development of science,
421.

Demonstration, the mania for, 18, 82 ;
artificial, 82.

Departure from free motion, 355.

Derived units, 278.

Descartes, on the measure of force,
148, 250, 270, 272-276 ; on quantity of
motion, conservation of momen-
tum, etc., 272 et seqq.; character of
his physical inquiries, 273 ; his me-
chanical ideas, 25a

Descent, on inclined planes, 134 et
seqq., law of, 137 ; in chords of cir-
cles, 138; vertical, motion of, treated
by Hamilton's principle,383; quick-
est, curve of. 426; ofcentre of grav-
ity. 52, 174 et seqq., 408.

Description, a fundamental feature of
science, 5.

Design evidences of, in nature, 452.

Determinants economy of, 487.

Determinative factor* of physical
processes 76.

Differences, of quantities their rOle
in nature, 236 ; of velocities, 325.

Differential calculus, 424.

Differential laws, 255, 461.

Dimensions, theory of, 279.

Dioptrics, Gauss's, economy of, 489.

Disillusionment, due to insight, 77.

DQhring, iz, 352.

Dynamics, the development of the
principles of, 128-235 '• retrospect of
the development of, 245-255; found-
ed by Galileo, 128 ; proposed new
foundations for, 243 ; chief results
of the development of, 245, 246;
analytical, founded by Lagrange on
the principle of virtual velocities,
467.

Earth, figure of, 395 et seqq.

Economical character of analytical
mechanics, 48a

Economy of description, 5.

Economy in nature, 459.

Economy of science, 481 -494.

Economy of thought, the basts and
essence of science, viii, 6, 481 ; of
language, 481 ; of all ideas, 482; of

526

THE SCIENCE OF MECHANICS.

the ideas cause and etfect, 484 ; of
the laws of nature, 485 ; of the law
of refraction, 485 ; of mathematics,
486; of determinants, 487 ; of cal-
culatinf( machines, 488 ; of Gauss's
dioptrics, moment of inertia, force-
function. 489.

Efflux, velocity of liquid, 402 et seq.

Egyptian monuments, i.

Eighteenth century, character of,
458.

Elastic bodies, 3x5, 317, 320.

Elastic rod, vibrations of. 490.

Elasticity, theory of, 258, 259, 490.

Electricity, revision of the theory of,
496.

Electromotor, Page's, 292 ; motion of
a free, 296 et seq.

Elementary laws, see Differential
laws.

Ellipsoid, trtaxal, 73 ; of inertia, 186;
central, 186.

Encyclopaedists, French, 463.

Energetics, the science of, 5x7.

Energy, Galileo's use of the word,
271 ; conservation of, 499 et seq.,
potential and kinetic, 272, 499; prin-
ciple of, 516 et seqq.

Enlightenment, the age of, 458.

Epstein, 511.

Equations, of motion, 34a; of me-
chanics, fundamental, 270.

Equilibrium, the decisive conditions
of, S3 ; dependence of, on a maxi-
mum or minimum of work, 69 ; sta-
ble, unstable, mixed, and neutral
equilibrium, 70-71 ; treated by
Gauss's principle, 355; figures of.
393; liquid, conditions of, 386 et
seqq.

Equipotential surfaces, see Level
surfaces.

Ergal, 499.

Error, our liability to in the recon-
struction of facts, 79.

Euler, on the " loi de repos," 68 ; on
moment of inertia, 179, 182, 186 ; on
the law of areas, 293 ; his form of
D'Alembert's principle, 337 ; on vis
viva, 348 : on the principle of least
action, 3G8 ; on the isoperimetrical

problems and the calculus of vajia-
tions, 433 et seqq.; bis theological
proclivities, 449, 455 ; his contribu-
tions to analytical mechanics, 466^

Exchange of velocities in impact, 315.

Experience, i et seq., 481, 49a

Explanation, 6.

Extravagance in nature, 4591

Facts and hypotheses, 494, 496, 498.

Fall of bodies, early views of, xaS;
investigation of the laws of. 130 et
seq.; see Descent.

Falling, sensation of, 906.

Falling bodies, laws of, accident of
their form, 247 et seq ; see Descent.

Faraday, 503 ; his lecture-experiment
on gases, 124.

Feelings, the attempt to explain them
by motions, 506.

Fermat, on the method of tai^ents,
423.

Fetishism, in modem ideas, 463.

Fiction of a boat in motion, Huy-
gens's. 315, 325.

Figure of the earth, 395 et seqq.

Films, liquid, 386, 392 et seq.

Flow, lines of, 400; of liquids, 4x6 et
seq.

Fluids, the principles of statics ap-
plied to, 86-110; see Liquids.

Fluid hypotheses, 496.

Force, moment of, 37 ; the experien-
tial nature of, 42-44 ; conception of,
in statics, 84 ; general attributes of,
8s ; the Galilean notion of, 142 ; dis-
pute concerning the measure of,
148, 250, 270, 274-276; centrifugal
and centripetal, is8 et seqq.; New-
ton on, 192, 197, 238, 239; moving,
203, 243 ; resident, impressed, cen-
tripetal, accelerative, moving. 338,
239 ; the Newtonian measure of, acq,
239t 276 ; lines of, 40a

Force-function, 398 et seqq., 479, 489;
Hamilton on, 3SO.

Force-relations, character of, 237.

Forces, the parallelogram of 3a, 33-48,
243; principle of the composition
and resolution of, 33-48, 197 et seq ;
triangle of, 108 ; mutual independ-

INDEX.

527

ence of, 154 ; livinKt see Vis viva ;
Newton on the paralleloKrain of,
X93i 197; impressed, equilibrated.ef-
fective, gained and lost, 336; mole-
cular, 384 et seqq.; functions of co-
ordinates, 397, 4C2 ; central, 397.

Formal development of science, 421.

Formulas, mechanical, 269-186.

Foucault and Toepler.optical method
of, 125.

Foucault' s pendulum, 302.

Fourier, on dimensions, 279.

Free rigid body, rotation of, 295.

Free systems, mutual action of, 287.

Friction, of minute bodies in liquids,
208 ; motion of liquids under, 4x6 et
seq.

Functions, mathematical, their office
in science, 492.

Fundamental equations of mechan-
ics, 27a

Funicular machine, 32.

Funnel, plunged in water, 412 ; rotat-
ing liquid in, 303.

"Galileo," name for unit of accel-
eration, 285.

Galileo, his dynamical achievements,
128-155; his deduction of the law
of the lever, 12; his explanation of
the inclined plane by the lever, 23 ;
his recognition of the principle of
virtual velocities, 51; his researches
in hydrostatics, 90; his theory of
the vacuum, 112 et seq.; his discov-
ery of the laws of falling bodies,
130 et seqq.; his clock^ 133; char-
acter of his inquiries, 140 ; his foun-
dation of the law of inertia, 143 ;
on the notion of acceleration, 145 ;
tabular presentment of his discov-
eries, 147 ; on the pendulum and the
motion of projectiles, 152 et seqq.;
founds dynamics, 128; his pendu-
lum, 162 ; his reasoning on the laws
of falling bodies, 230, 131, 247 ; his
favorite concepts, 250 ; on impact,
308-312; his struggle with the
church, 446 ; on the strength of ma-
terials, 451; does not mingle science
with theology, 457 ; on inertia, 509.

Gaseous bodies, the principles of
statics applied to, 1 10-127.

Gases, flow of, 405 ; compression of,
407.

Gauss, his view of the principle of
virtual velocities, 76 ; on absolute
units, 278; his principle of least
constraint, 350-364; on the statics
of liquids, J90; his dioptrics, 489.

Gilbert, 462.

Grassi, 94.

Grassmann, 480.

Gravitation, universal, 190.

Gravitational system of measures,
284-286U

Gravity, centre of. See Centre of
gravity.

Green' s Theorem, 109.

Guericke, his theological specula-
tions, 448 ; his experiments in aero-
statics, 117 et seqq.; his notion of
air, 1x8; his air-pump, x2o; his air-
gun, 123.

Gyration, centre of, 334.

Halley, 448.

Hamilton, on force-function, 350.

Hamilton's principle, 380-384, 48a

Heat, revision of the theory of, 496.

Helm, 517.

Hehnholts, viii; on the conservation
of energy, 499, 501, 518.

Hemispheres, the Magdeburg, 122.

Hermann, emplojrs a form of D' Alem-
bert's principle, 337; on motion in
a resisting medium, 435.

Hero, his fountain, 4x1 ; on the mo-
tion of light, 422 ; on maxima and
minima, 451.

Hiero, 86.

Hipp, chronoscope of, 151.

Hollow space, liquids enclosing, 392.

Homogenous, 279.

Hopital, L*, on the centre of oscilla-
tion, 33X ; on the brachistochrone,
4^6.

Horror vaeui^ xi2.

Hume, on causality, 484.

Huygens,dynamical achievements of,
155- X87; his deduction of the law
of the lever, X5-16; criticism of his

528

THE SCIENCE OF MECHANICS.

deduction, 17-18; his rank aa an
inquirer, 155 ; character of hta re-
aearchea, 156 et seq.; on centrifugal
and ceniripetal force, 158 et seqq.;
his experiment with light balls in
rotating fluids, 162; on the pendu-
lum and oscillatory motion, i6a et
seqq.; on the centre of oscillation,
173 et seq.; his principle of the de-
scent and rise of the centre of grav-
ity, 274 ; his lesser investigations,
x86; his crowning achievement, 187;
his favorite concepts, aji ; on im-
pact, 313-327; on the principle of
visviva^ 343, 348; on the figure of
the earth, 395 ; his optical re-
searches, 423 ; does not mingle sci-
ence and theology, 457.

Hydraulic ram, Montgolfier's, 41X.

Hydrodynamic pressure, 413.

Hydrodynamics, 402-420.

Hydrostatic pressure, 413.

Hydrostatics, 384-402.

Hypotheses and facts, 494.

Inclined plane, the principle of the,
24-33; Galileo's deduction of its
laws, 131 ; descent on, 354 ; movable
on rollers, 357 et seq.

Indeterminate coefficients, La-
grange's, 471 et seq.

Inelastic bodies, 317, 318.

Inertia, history of the law of, 141, 143,
509, SIX et seqq.; moment of, X79,
182, x86, 489 ; bodies with variable
moments of, 302 ; law of, critically
elucidated, 232, 238; Newton on,
238, 243.

Inertial system, 515.

Impact, the laws of, 305-330; force of.
compared with pressure, 3x2; in
the Newtonian view, 317 et seqq.;
oblique, 327; Maupertuis's treat-
ment of, 365.

Impetus, 275.

Impulse, 27X.

Inquirers, the great, character and
value of their performances, 7;
their different tasks, 76 ; their atti-
tude towards religion, 457.

Inquiry, typical modes of, 3x7.

Instinct, mechanical, importance of,
304.

Instinctive knowledge, its cogenqr.
origin, and character, x, 26-2&, 83.

Instincts, our animal, 463.

Instruction, various methods of, 5.

Integral laws, 255, 46X.

Intelligence, conception of, in nature,
461.

Interdependence of the facts of na-
ture, 502 et passim.

Internal forces, action of, on free srs-
leins, 289, 295.

International language, 48X.

Isoperimetrical problems, 42x1446;
Euler's classification of, 433.

Isothermal surfaces, 400.

Jacobi, 76, 38X, 459; on principle of

least action, 371.
Jellett, on the calculus of variations,

437 et seq.
Jolly, iz.
Joule, 50X.
Judgments, economical character of

all, 483.

Kant, on causality, 484.

Kater, 186.

Kepler, his laws of planetary motion,
X87 ; possibility of his discovery of
the laws of falling bodies, 248; on
maxima and minima, 423 ; on astrol-
ogy, 463.

Kilogramme, 28x.

Kilogramme-metre, 272.

Kinetic energy, 272, 499.

Kirchhoff, viii, 38X.

Knowledge, instinctive, x, 26-28, 85;
the communication of, the founda-
tion of science, 4 ; the nature of, S \
the necessary and sufficient condi-
tions of, 10.

KOnig, on the cells of the honeycomb,

453-

Laborde, apparatus of, xja

Lagrange, his deduction of the law of
the lever, 13; his deduction of the
principle of virtual velocities, 65-
67 ; criticism of this last deductioHi

INDEX.

529

67-68; his form of D'Alembert's
principle, 337 ; on vis viva, 349 ; on
the principle of least action, 371 ;
on the calculus of variations, 436 et
seq.; emancipates physics from the-
ology, 457; his analytical mechan-
ics, ix, 466; his indeterminate co-
efficients. 471 et seq.
Lami, on the composition of forces,

36.

Lange 51X et seq.

Language, economical character of,
481 ; possibility of a universal, 48a ;
the Chinese, 482.

Laplace, 463.

Lateral pressure, 103.

Laws of nature, 502.

Laws, rules for the mental recon-
struction of facts, 83-84, 485.

Least action, principle of, 364-380;
its theological kernel, 454.

Least constraint, principle of, 350-

364.

Leibnitz, on the measure of force,
148. ajo, 270. 274-276 ; on quantity of
motion, 274 ; on the motion of light,
425, 454; on the brachistochrone,
426 ; as a theologian, 449.

Level surfaces, 98, 398 et seqq.

Lever, the principle of the, 8-25 ; " po-
tential, ' 20 ; application of its prin-
ciples to the explanation of the
other machines, 22; its law deduced
by Newton's principles, 263-^267;
conditions of its rigidity, 96; Mau-
pertuis's treatment of, 366.

Libraries, stored up experience, 481.

Light, motion of, 422, 424, 426; Mau-
pertuis on motion of, 367 ; motion
of, in refracting media, 374-376, 377-
379 ; its minimal action explained,

459-
LindelOf, 437.

Lippich, apparatus of, xja

Liquid efflux, velocity of, 402.

Liquid-head, 403, 416.

Liquid, rotating in a funnel, 303.

Liquids, the statics of, 86-izo; the
dynamics of, 402-420 ; fundamental
properties of, 91 ; compressibility
of, 92 ; equilibrium of, subjected to

gravity, 96; immersed in liquids,
pressure of, 105 ; lateral pressure of,
103 ; weightless, 384 et seqq.; com-
pression of, 407 ; soniferous, vibra-
tions of, 407 ; mobile, 407 ; motion
of viscous, 416.

Living power, 272.

Lobatschewsky, 493.

Locomotive, oscillations of the body
of, 292.

Luther, 463.

Machines, the simple, 8 et seqq.

Maclaurin on the cells of the honey-
comb, 453; his contributions to ana-
lytical mechanics, 466.

Magnus, Valerianus, 1x7.

Manometer, statical, 123.

Maraldi, on the honeycomb, 453.

Marci, Marcus, 305-308.

Mariotte, his law, X2S ; his apparatus
and experiments, 126 et seq.; on im-
pact, 313.

Mass-areas, 295.

Mass, criticism of the concept of, 216
-222 ; Newton on, 192, 194, 217, 238,
251 ; John Bernoulli on, 251 ; as a
physical property, 194; distin-
guished from weight, 195; measura-
ble by weight, 195, 220; scientific
definition of, 218 et seq., 243, 510;
involves principle of reaction, 220.

Mass, motion of a, in principle of
least action, 372.

Mathematics, function of, 77.

Matter, quantity of, 216, 238.

Maupertuis, his loi dt re^s, 68 et
seq.; on the principle of least ac-
tion, 364, 368; his theological pro-
clivities, 454.

Maxima and minima, 368 et seqq.
problems of, 422 et seqq.

Maximal and minimal effects, ex
planation of, 460.

Maxims, scholastic, 143.

Maxwell, 271.

Mayer, J. R., on work, 249, 503, 518;
his physical achievements, 501.

Measures, see Units.

Mechanical, experiences, i ; knowl-
edge of antiquity, 1-3; phenomena.

530

THE SCIENCE OF MECHANICS.

purely, 493 et seq.: theory of na-
ture, its untenability, 49s et seq.;
phenomena not fundamental, 496 ;
conception of the world, artificiality
of, 496.

Mechanics, the science of, i; earliest
researches in, 8 ; extended applica-
tion of the principles of, and de-
ductivedevelopment of the science,
255-420; the formula and units of.
269-286; character of the principles
of, 237; form of its principles
mainly of historical and accidental
origin, 247 et seq.; tfieological, ani-
mistic, and mystical points of view
in, 446-465 ; fundamental equations
of, 270-276 ; new transformation of,
4?o; relations of, to other depart-
ments of knowledge, 495-307 ; rela-
tions of, to physics, 495-504; rela-
tions of, to physiology, 504-507; an
aspect, not the foundation of the
world, 496, 3'7 ; analytical, 465-480;
Newton's geometrical, 465.

Mcdiinn, motion-determinative, hy-
pothesis of, in space, 230; resisting,
motion in, 435.

Memory, 481, 488.

Mensbrugghe, Van der, on liquid
films, 386.

Mental arti fices, 492 et seqq.

Mercurial air-pump, 125.

Merscnne, Z14, 174.

Method of tangents, 423.

Metre, 280.

Mimicking, of facts in thought, see
Reproduction.

Minima, see Maxima.

Minimum of superficial area, 387.

Mixed equilibrium, 70-71.

Mobile liquids, 407.

MObius, 372, 480.

Models, mental, 492.

Molecular forces, 384 et seqq.

Moment, statical, 14 ; of force, 37; of
inertia, 179, 182. \^\

Moments, virtual, 57.

Momentum, 241, 244. 271 ; law of the
conservation of, 288 ; conservation
of, interpreted, 326.

Monistic philosophy, the, 465.

Montgolfier's hydranlic ram, 411.

Moon, its acceleration towards the
earth, 190; length of its day in-
creased to a month, 299.

Morin, apparatus of, 150L

Motion, Newton's laws of, 227,241;
quantity of, 238, 271 et seqq.; equa-
tions of. 342, 371 ; circular, laws of,
158 et seqq.; uniformly accelerated,
132; relative and absolute, %rj et
seqq.. 511 et seq.

Motivation, law of, 484.

MQller, F. A., 517.

Mystical points of view in mechanics,

456.
Mysticism in science. 482.
Mythology, mechanical. 464.

Napier, his theological inclinations,

447.
Nature, laws of, 503.

Necessity, 484, 485.

Neumann, C, 355.

Neutral equilibrium, 70-71.

Newton,his dynamical achievements,
187-201 ; his views of time, space,
and motion, 222-238; synoptical
critique of his enunciations, 2jS-
245; scope of his principles, 256-269;
enunciates the principle of the par-
allelogram of forces, 36 ; his prin-
ciple of similitude, 165 et seq.; his
discovery of universal gravitation,
its character, and its law, x88 et
seqq.; effect of this discovery on
mechanics, 191 ; his mechanical
discoveries 192 ; his rtgultK philo-
tophandi^ 193; his idea of force,
193 ; his concept of mass, 194 et
seqq.; on the composition of forces,
197; on action and reaction, 198;
defects and merits of his doctrines,
20X, 244 ; on the tides, 209 et seq-;
his definitions, laws, and corolla-
ries, 238-242 ; his water-pendulum,
409; his theological speculations,
448 ; the economy and wealth of bis
ideas, 269 ; his laws and definitions,
proposed substitutes for, 243; his
favorite concepts, 251 ; on the figure
of the earth, 395 ; does not mingle

INDEX,

531

theology with science, 457 ; on the
brachistochrone, 436.
Numbers, 486.

Observation, 82.

Occasionalism, the doctrine of, 449.

Oersted, 93.

Oil, use of, in Plateau's experiments,

384 et seq.
Oscillation, centre of, 331-335.
Oscillatory motion, 162 et seqq.

Pagan ideas in modem life, 462.

Page' s electromotor, 992.

Pappus, 422 ; on maxima ard min-
ima. 4SI.

Parallelism of strata, 409.

Parallelogram of forces, see Forces.

Particular determination, principle
of, 5x3-

Pascal, his application of the prin-
ciple of virtual velocities to the
statics of liquids, 54, 91, 96 ; his ex-
periments in liquid pressure, 99;
his paradox, ioz-102 ; his great pi-
ety, 447 ; criticism of his deduction
of the hydrostatic principle, 95-96 ;
his experiments in atmospheric
pressure. Z14 et seqq.

Peltier's effect, 503.

Pendulum, motion of, 153, x<^, x68;
law of motion of, 168 ; experiments
illustrative of motion of, 168 et
seqq.; conical, 171; determination
of ^ by, 172 ; simple and compound,
173* 'm\ cycloidal, 186; a falling,
205; ballistic, 328 ; liquid, 409.

Percussion, see Impact; centre of,

327.
Percussion-machine, 3x3.
Perier, 115.

Perpetual motion, 25, 89, 500.
Philosophy of the specialist, the, 506.
Phoronomic similarity, 166.
Physics and theology, separation of,

456.

Physics, arti6cial division of, 495 ;
necessity of a comparative, 498 ; re-
lations of mechanics to. 495-504 ;
disproportionate formal develop-
ment of, 505.

Physiology, relations of mechanics

to, 504-507; distinguished from
physics, 507.

Pila Heronis, 118,4x8.

Place, 222, 226.

Planck, 517.

Planets, motion of, 187 et seq.

Plateau, on the statics of liquids, 384
'394*1 Plateau's problem, 393.

Poggendorfs apparatus, 206 et seq.

Poinsot, x86, 25X, 269, 480.

Poisson, 42, 46.

Polar and parallel coordinates, 304.

Poncelet, 251, 272.

Popper, J., 5x6.

Porta, 462.

Poske, on the law of inertia, 509.

Potential, xxo, 398 et seqq.; potential
function, 497 ; potential energy, 499.

Pound, Imperial, Troy, Avoirdupois,
283.

Pre-established harmony, 449.

Pressure, origin of the notion of, 84 \
liquid, 90,99 et seqq.; of falling bod-
ies, 205 ; hydrodynamic and hydro-
static, 4x3; of liquids in motion,

414.
Pressure-bead, 403, 4x6.

Principles, their general character
and accidental form, 79, 83, 431 ; see
Laws.

Projectiles, motion of, 152 et seqq.;
treated by the principle of least ac-
tion, 369.

Projection, obllqae, 153; range of,

154.
Proof, the natural methods of, 8a
Ptolemy, 332,
Pulleys, 81, 49-51*
Pump, 1X3.
Pythagoras, 433.

Quantity, of matter, 2x6, 238 ; of mo-
tion. 238, 37X et seqq.
Quickest descent, curve of, 436.

Radii vteiores^ 394.

Rationalism, 458.

Reaction, discussion and illoatration
of the principle of, 201-4x6; criti-
cism of the principle of, 3x6-333;
Newton on, 196, 20X, 242.

Reaction-tubes, 301.

532

THE SCIENCE OF MECHANICS.

Reaction-wheels, 399 et seqq.

R^atimnr, 453.

Reason, sufficient, principle of, 9, 484,

50a,
Reconstruction of facts, mental, see

Reproduction.
RefiKurinK of facts in thought, see

Reproduction.
Refraction, economical character of

law of, 485.
Regultt Philoso^handi, Newton's, 193.
Regularity, 395.
Religious opinions, Our, 464.
Repos, hide, 68.
Representation, see Reproduction of

facts in thought.
Reproduction of facts in thought, 5,

84, 421. 481 494.
Research, means and aims of, distin-
guished, 507.
Resistance head, 417.
Rest, Maupertuis's law of, 68, 259.
Resultant of force, 34.
Richer, 161, 251.
Riemann, 493.
Roberval, his balance, 60; his method

of maxima and minima. 423: on

momenta, 305 ; on the composition

of forces, 197.
Robins, 330.
Routh, 352.

Routine methods, 181, 268, 287, 341,
Rules, 83. 483; the testing of, 81.

Sail filled with wind, curve of, 431.

Scheffler, 353, 364.

Schopenhauer, on causality, 484.

Science, the nature and development

■ of, 1-7 ; the origin of, 4, 8, 78 ; de-
ductive and formal development of,
421 ; physical, its pretensions and
attitude 464 et seq.; the economy
of, 481 494 ; a minimal problem,
490 ; the object of, 496, 497, 502. 507;
means and aims of. should be dis-
tinguished, 504, 505 ; condition of
the true development of, 504 ; divi-
sion of labor in, 505 ; tools and in-
struments of, 505.

Science and theology, conflict of, 446;
their points of identity, 460.

Scientists, struggle of, with their own
preconceived ideas, 447.

Seebeck's phenomenon, 503.

Segner, 186; Segncr's wheel, 309.

Sensations, analysis of, 464 ; the ele-
ments of nature, 482 ; their relative
realness, 506.

Shortest line, 369, 371.

Similarity, phoronomic, 166.

Similitude, the principle of, 166, 177.

Siphon, 114 et seqq.

Space, Newton on, 226; absolute and
relative, 226, 232; a set of sensa-
tions, 506; multi-dimensioned, an
artifice of thought, 493.

Spannkraft, 499.

Specific gravity, 87-88.

Sphere, rolling on inclined plane,

346.

Spiritism, or spiritualism, 49

Stable equilibrium, 70-71.

Stage of thought, the, 505.

Statical manometer, 123.

Statical moment, 14 ; possible origin
of the idea, 21.

Statics, deduction of its principles
from hydrostatics, 107 et seqq.; the
development of the principles of,
8 127 ; retrospect of the develop-
ment of, 77 85 ; the principles of»
applied to fluids, 86 no; the prin-
ciples of,applied to gaseous bodies,
no 127; Varignon's dynamical, 38,
368 ; analytical, founded by La-
grange on the principle of virtual
velocities, 467.

Stevinus, his deduction of the law of
the inclined plane, 24-31 ; his ex-
planation of the other machines by
thc inclined plane, 31-33; the par-
allelogram of forces derived from
his principle, 32-35 ; his discovery
of the germ of the principle of vir-
tual velocities,49-5i; his researches
in hydrostatics, 88-90; his broad
view of nature, 500.

Strata, parallelism of, 409.

Streintz, 510 et seqq.

String, equilibrium of a, 372 et seqq.;
see Catenary.

Strings, equilibrium of three>knotr

INDEX.

533

ted, 6i ; equilibrium of ramifying,

33-
Suction, 113.
Sufficient reason, the principle of, 9,

484, 502.
Surface of liquids, connection of, with

equilibrium, 386-390.
Surfaces, isothermal, 400; level, 98,

398 et seqq.
Symmetry of liquid films explained,

394.
Synoptical critique of the Newtonian

enunciations. 238-245.
Synthetic method, 466.

Tangents, method of. 423.

Taylor, Brook, on the centre of os-
cillation, 335.

Teleology, or evidences of design in
nature. 452.

Theological points of view in me-
chanics, 446 et seqq.; inclinations
of great physicists, 450.

Theology and science, conflict of, 446;
their points of identity, 460.

Theorems, 421.

Theories. 491 et seqq.

Thermometers, their construction,
282.

Things, their nature, 482 ; things of
thought, 492 et seqq.

Thomson and Tait. their opinion of
Newton's laws, 245.

Thought, instruments of, 505 ; things
of, 492 et seqq.; economy of, see
Economy.

Tides, Newton on, 209 et seq.; their
effect on the armyof Alexander the
Great, 909; explanation of, 213 et
seq.; their action illustrated by an
experiment, 215.

Time, 511; a set of sensations, 506;
Newton's view of, 222-^38; abso-
lute and relative, 222; nature of,
223-226, 234.

Toeppler and Foucault, optical
method of, 225.

Torricclli, his modification of Gali-
leo's deduction of the law of the
inclined plane, 52; his measure-
ment of the weight of the atmos-

phere, 113; founds dynamics, 402 ;
his vacuum experiment, 113; founds
hydrodynamics, 402; on the velo-
city of liquid efflux, 402.

Trade winds, 302.

Trades and classes, function of, in the
development of science, 4.

Tubes, motion of liquids in, 416 et
seqq.

Tylor, 462, 463.

Ubaldi, Guido, his statical re-
searches, 21.
Uniquely determined, zo, 502.
Unitary conception of nature, 5.
Units. 269-286.
Unstable equilibrium, 70-71.

Vacuum, 112 et seqq.

Variation, of curve-elements, 433 et
seqq.

Variations, calculus of, 436 et seqq.

Varignon, enunciates the principle of
the parallelogram of forces, 36; 011
the simple machines, 37 ; his statics
a dynamical statics, 38; on velocity
of liquid efflux, 403.

Vas tM^rficiaritiin of Stevinus, 89.

Vehicle on wheels, 291.

Velocity, 144; angular, 396; a phys-
ical level, 325.

Velocity-head, 4x7.

Vibration, see Oscillation.

View, breadth of, possessed by all
great inquirers. 500 et seq.

Vinci, Leonardo Da, on the law of
the lever, 30.

Virtual displacements, definition of,
37 ; see also Virtual velocities.

Virtual moments, 57.

Virtual velocities, origin and mean-
ing of the term, 49; the principle
of, 49-77.

Viscosity of liquids, 416. *

Vii mortua^ 272, 275.

Vii viva, 272 et seqq., 315 ; conserva-
tion of, 317, interpreted, 326; in im-
pact, 323 et seqq.; principle of, 343-
350 ; connection of Huygens' s prin^
ciple with, 178; principle of. de-
duced from Lagrange's fundamen-
tal equations, 478, 4S)9.

534

THE SCIENCE OF MEClfANICS.

Vitnivius, on the nature of sound, 3 ;
his accouut of Archimedes' s dis-
covery, 86; on ancient air-instru-
ments, xia

Viviani, 113.

Voltaire, 449, 454.

Volume of liquids, connection of
with equilibrium, 387-390.

Wallis, on impact, 313 ; on the centre

of percussion, 327.
Water, compressibility of, 93.
Weightless liquids, 384 et seqq.
Weights and measures, see Units.
Weston, differential pulley of, 59.
Whcatstone, chronoscope of, 151.
Wheel and axle, with non-circular

wheel, 72 ; motion of, 22 et seq., 60,

337. 344. 354. 381.
Will, conception of, in nature, 461.
Wire frames, Plateau's, 393.

Wohlwill, on the law of inertia, 306.

509.
Wood, on the cells of the honeycomb,

453.

Woodhouse, on isoperimetrical prob-
lems, 430.

Work, 54, 67 et seq., 248 et seq., 363;
definition of, 272; determinative of
vi» viva^ 178; accidentally not the
original concept of mechanics, 54S:
J. R. Mayer's views of, 249; Hay-
gens' s appreciation of. 252, 27a ; in
impact, 322 et seqq.; of molecular
forces in liquids, 385 et seqq.; posi-
tive and negative, 386; of liquid
forces of pressure, 415; of com-
pression, 407.

Wren, on impact, 313.

Wright, Chauncey, 453.

Yard, Imperial, 281; American. 283.

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